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ENGINEERING 
DESCRIPTIVE  GEOMETRY 


^Engineering 
Descriptive  Geometry 


A  Treatise  on  Descriptive  Geometry  as  the  Basis  of  Mechanical 

Drawing,  Explaining  Geometrically  the  Operations 

Customary  in  the  Draughting  Room 


BY 

P.  W.  BARTLETT 

commandIer,  u.  s.  navy 

HEAD  OF  DEPARTMENT  OF  MARINE  ENGINEERING  AND  NAVAL  CONSTRUCTION 
AT  THE  UNITED  STATES  NAVAL  ACADEMY 

AND 

THEODORE  W.  JOHNSON 

A.  B.,   M.E. 

PROFESSOR  OF  MECHANICAL  DRAWING,  UNITED  STATES  NAVAL  ACADEMY 
MEMBER  OF  AMERICAN  SOCIETY  OF  MECHANICAL  ENGINEERS 


or  THf 
UNIVERSITY 

OF 


NEW   YORK 
JOHN    WILEY    &    SONS 

London  :  CHAPMAN  &  HALL,  Limited 
1910 


1^3 


fi%/Mt 


^c 


Copyright,  1910,  by 
T.  W.  Johnson 


BALTIMORE,  MD.,  V.  S.  A. 


'^     OF   THE 

UNIVERSITY 

OF 


PEEFACE. 


The  aim  of  this  work  is  to  make  Descriptive  Geometry  an 
integral  part  of  a  course  in  Mechanical  or  Engineering  Drawing. 

The  older  books  on  Descriptive  Geometry  are  geometrical  rather 
than'  descriptive.  Their  authors  were  interested  in  the  subject  as  a 
branch  of  mathematics,  not  as  a  branch  of  drawing. 

Technical  schools  should  aim  to  produce  engineers  rather  than 
mathematicians,  and  the  subject  is  here  presented  with  the  idea 
that  it  may  fit  naturally  in  a  general  course  in  Mechanical  Drawing. 
It  should  follow  that  portion  of  Mechanical  Drawing  called  Line 
Drawing,  whose  aim  is  to  teach  the  handling  of  the  drawing  instru- 
ments, and  sliould  precede  courses  specializing  in  the  various 
branches  of  drawing,  such  as  Mechanical,  Structural,  Architectural, 
and  Topographical  Drawing,  or  the  "  Laying  Off  "  of  ship  lines. 

The  various  branches  of  drawing  used  in  the  different  industries 
may  be  regarded  as  dialects  of  a  common  language.  A  drawing  is 
but  a  written  page  conveying  by  the  use  of  lines  a  mass  of  informa- 
tion about  the  geometrical  shapes  of  objects  impossible  to  describe 
in  words  without  tedium  and  ambiguity.  In  a  broad  sense  all  these 
branches  come  under  the  general  term  Descriptive  Geometry.  It 
is  more  usual,  however,  tp  speak  of  them  as  branches  of  Engineer- 
ing Drawing,  and  that  term  may  well  be  used  as  the  broad  label.. 

The  term  descriptive  Geometry  will  be  restricted,  therefore,  to- 
the  common  geometrical  basis  or  ground  work  on  which  the  various 
industrial  branches  rest.  This  ground  work  of  mathematical  laws, 
is  unchanging  and  permanent. 

The  branches  of  Engineering  Drawing  have  each  their  own 
abbreviations  and  special  methods  adapting  them  to  their  own 
particular  fields,  and  these- conventional  methods  change  from  time 
to  time,  keeping  pace  with  changing  industrial  methods. 

Descriptive  Geometry,  though  unchanged  in  its  principles,  has 
recently  undergone  a  complete  change  in  point  of  view.  In 
changing  its  purpose  from  a  matliematical  one  to  a  descriptive  one, 
or,  from  being  a  training  for  the  geometrical  powers  of  a  mathema- 
tician to  being  a  foundation  on  which  to  build  up  a  knowleclge  of 


203782 


Ti  Preface 

some  branch  of  Engineering  Drawing,  the  number  and  position  of 
the  planes  of  projection  commonly  used  are  altered.  The  object  is 
now  placed  behind  the  planes  of  projection  instead  of  in  front  of 
them,  a  change  often  spoken  of  as  a  change  from  the  "  1st  quad- 
rant "  to  the  "  3d  quadrant,"  or  from  the  French  to  the  American 
method.  We  make  this  change,  regarding  the  3d  quadrant  method 
as  the  only  natural  method  for  American  engineers.  All  tlie  prin- 
ciples of  Descriptive  Geometry  are  as  true  for  one  method  as  for  the 
other,  and  the  industrial  branches,  as  Mechanical  Drawing,  Struc- 
tural Drawing,  etc.,  as  practiced  in  this  country,  all  demand  this 
method. 

In  addition,  the  older  geometries  made  practically  no  use  of  a 
third  plane  of  projection,  and  we  take  in  this  book  the  further  step 
of  regarding  the  use  of  three  planes  of  projection  as  the  rule,  not 
the  exception.  To  meet  the  common  practice  in  industrial  branches, 
we  use  as  our  most  prominent  metliod  of  treatment,  or  tool,  the 
auxiliary  plane  of  projection,  a  device  which  may  be  called  the 
draftsman's  favorite  method,  but  which  in  books  is  very  little 
noticed. 

As  the  work  is  intended  for  students  who  are  but  just  taking  up 
geometry  of  three  dimensions,  in  order  to  inculcate  by  degrees  a 
power  of  visualizing  in  space,  we  begin  the  subject,  not  with  the 
mathematical  point  in  space  but  with  a  solid  tangible  object  shown 
by  a  perspective  drawing.  No  exact  construction  is  based  on  the 
perspective  drawings  which  are  freely  used  to  make  a  realistic  ap- 
pearance. As  soon  as  the  student  has  grasped  the  idea  of  what 
orthographic  projection  is,  knowledge  of  how  to  make  the  projection 
is  taught  by  the  constructive  process,  beginning  with  the  point  and 
passing  through  the  line  to  the  plane.  To  make  the  subject  as 
tangible  as  possible,  the  finite  straight  line  and  the  finite  portion  of 
a  plane  take  precedence  over  the  infinite  line  and  plane.  These 
latter  require  higher  powers  of  space  imagination,  and  are  therefore 
postponed  until  the  student  has  had  time  to  acquire  such  powers 
from  the  more  naturally  understood  branches  of  the  subject. 

F.  W.  B. 
T.  W.  J. 

Mabch,  1910. 


CONTENTS. 

CHAPTER  PAGE 

I.     Nature  of  Orthographic  Projection 1 

II.     Orthographic  Projection  of  the  Finite  Straight  Line.  ..  18 

III.  The  True  Length  of  a  Line  in  Space 27 

IV.  Plane   Surfaces   and  Their   Intersections   and   Develop- 

ments      38 

V.     Curved  Lines   49 

VI.     Curved  Surfaces  and  Their  Elements 62 

VII.     Intersections  of  Curved   Surfaces 70 

VIII.     Intersections  of  Curved  Surfaces;    Continued 81 

IX.     Development  of  Curved  Surfaces 91 

X.     Straight  Lines  of  Unlimited  Length  and  Their  Traces.  .  98 

XI.     Planes  of  Unlimited  Extent:  Their  Traces 108 

XII.     Various  Applications   121 

XIII.  The  Elements  of  Isometric  Sketching 133 

XIV.  Isometric  Drawing  as  an  Exact  System 142 

Set  of  Descriptive  Drawings 149 


OF  THE 

UNIVERSITY 

OF 

=£d^lFORH^ 


CHAPTER  I. 
NATURE  OF  ORTHOGRAPHIC  PROJECTION. 

1.  Orthographic  Projection. — The  object  of  Mechanical  Draw- 
ing is  to  represent  solids  with  such  mathematical  accuracy  and 
precision  that  from  the  drawing  alone  the  object  can  be  built  or 
constructed  without  deviating  in  the  slightest  from  the  intended 
shape.  As  a  consequence  the  "  working  drawing "  is  the  ideal 
sought  for,  and  any  attempt  at  artistic  or  striking  effects  as  in 
"  show  drawings ''  must  be  regarded  purely  as  a  side  issue  of  minor 
importance.  .  Indeed  mechanical  drawing  does  not  even  aim  to 
give  a  picture  of  the  object  as  it  appears  in  nature,  but  the  views 
are  drawn  for  the  mind,  not  the  eye. 

The  shapes  used  in  machinery  are  bounded  by  surfaces  of  mathe- 
matical regularity,  such  as  planes,  cylinders,  cones;  and  surfaces 
of  revolution.  They  are  not  random  surfaces  like  the  surface  of  a 
lump  of  putty  or  other  surfaces  called  "  shapeless."  These  definite 
shapes  must  be  represented  on  the  flat  surface  of  the  paper  in  an 
unmistakable  manner. 

The  method  chosen  is  that  known  as  *'*'  orthographic  projection/* 
If  a  plane  is  imagined  to  be  situated  in  front  of  an  object,  and 
from  any  salient  point,  an  edge  or  corner,  a  perpendicular  line, 
called  a  "projector,"  is  drawn  to  the  plane,  this  line  is  said  to 
project  the  given  point  upon  the  plane,  and  the  foot  of  this  perpen- 
dicular line  is  called  the  projection  of  the  given  point.  If  all 
salient  points  are  projected  by  this  method,  the  orthographic  draw- 
ing of  the  object  is  formed. 

2.  Perspective  Drawing. — The  views  we  are  accustomed  to  in 
artistic  and  photographic  representations  are  "  Perspective  Views." 
They  seek  to  represent  objects  exactly  as  they  appear  in  nature. 
In  their  case  a  plane  is  supposed  to  be  erected  between  the  human 
eye  and  the  object,  and  the  image  is  formed  on  the  plane  by  sup- 
posing straight  lines  drawn  from  the  eye  to  all  salient  points  of 


Engineerixg  Descriptive  Geometry 


the  object.  Where  these  lines  from  the  eye,  or  "  Visual  Rays/'  as 
they  are  called,  pierce  the  plane,  the  image  is  formed. 

Fig.  1  represents  the  two  contrasted  methods  applied  to  a  simple 
object,  and  the  customary  nomenclature. 

An  orthographic  view  is  sometimes  called  an  "  Infinite  Perspec- 
tive View,''  as  it  is  the  view  which  could  only  be  seen  by  an  eye  at 
an  infinite  distance  from  the  object.  "  The  Projectors  "  may  then 
be  considered  as  parallel  visual  rays  which  meet  at  infinity,  where 
the  eye  of  the  observer  is  imagined  to  be. 


Projector 


Perspectjve  View. 


Orthographic  View. 


Fig.  1. 


3.  The  Regular  Orthographic  Views. — Since  solids  have  three 
"  dimensions,"  length,  breadth  and  thickness,  and  the  plane  of  the 
paper  on  which  the  drawing  is  made  has  but  two,  a  single  ortho- 
graphic view  can  express  two  only  of  the  thrjee  dimensions  of  the 
object,  but  must  always  leave  one  indefinite.  Points  and  lines  at 
different  distances  from  the  eye  are  drawn  as  if  h^ing  in  the  same 
plane.  From  one  view  only  the  mind  can  imagine  them  at  dif- 
ferent distances  by  a  kind  of  guess-work.  If  two  views  are  made 
from  different  positions,  each  view  may  supplement  the  other  in 
the  features  in  which  it  is  lacking,  and  so  render  the  representa- 
tion entirely  exact.  Theoretically  two  views  are  always  required 
to  represent  a  solid  accurately. 

To  make  a  drawing  all  the  more  clear,  other  views  are  generally 
advisable,  and  three  views  may  be  taken  as  the  average  requirement 
for  single  pieces  of  machinery.  Six  regular  views  are  possible, 
however,  and  an  endless  number  of  auxiliary  views  and  "  sections  " 
in  addition.  For  the  present,  we  shall  consider  only  the  "  regular 
views,"  which  are  six  in  number. 


Nature  of  Orthographic  Projection 


4.  Planes  of  Projection. — A  solid  object  to  be  represented  is 
supposed  to  be  surrounded  by  planes  at  short  distances  from  it,  the 
planes  being  perpendicular  to  each  other.  From  each  point  of 
every  salient  edge  of  the  object,  lines  are  supposed  to  be  drawn 
perpendicular  to  each  of  the  surrounding  planes,  and  the  succes- 
sion of  points  where  these  imaginary  projecting  lines  cut  the  planes 
are  supposed  to  form  the  lines  of  the  drawings  on  these  planes. 
One  of  the  planes  is  chosen  for  the  plane  of  the  paper  of  the  actual 
drawing.  To  bring  the  others  into  coincidence  with  it,  so  as  to 
have  all  of  them  on  one  flat  sheet,  they  are  imagined  to  be  unfolded 
from  about  the  object  by  revolving  them  about  their  lines  of  inter- 
section with  each  other.  These  lines  of  intersection,  called  "  axes 
of  projection,"  separate  the  flat  drawing  into  different  views  or 
elevations. 


Fia.  2. 


Engineering  Descriptive  Geometry 


Fig.  2a. 


Fig.  2  is  a  true  perspective  drawing  of  a  solid  object  and  the 
planes  as  they  are  supposed  to  surround  it.  This  figure  is  not  a 
mechanical  drawing,  but  represents  the  mental  process  by  which 
the  mechanical  drawing  is  supposed  to  be  formed  by  the  projection 
of  the  views  on  the  planes.  In  this  case  the  planes  are  supposed 
to  be  in  the  form  of  a  perfect  cube.  The  top  face  of  the  cube  shows 
the  drawing  on  that  face  projected  from  the  solid  by  fine  dotted 
lines.  Eemember  that  these  fine  dotted  lines  are  supposed  to  be 
perpendicular  to  the  top  plane.  This  drawing  on  the  top  plane  is 
called  the  "  plan."  On  the  front  of  the  cube  the  "  front  view  "  or 
"  front  elevation  '^  is  drawn,  and  on  the  right  side  of  the  cube  is 


Nature  of  Orthographic  Projection  5 

the  "  right  side  elevation/^  Three  other  views  are  supposed  to  be 
drawn  on  the  other  faces  of  the  cube,  but  they  are  shown  on  Fig. 
2a,  which  is  the  perspective  view  of  the  cube  from  the  opposite 
point  of  view,  that  is,  from  the  back  and  from  below  instead  of 
from  in  front  and  from  above. 

This  method  of  putting  the  object  to  be  drawn  in  the  center  of 
a  cube  of  transparent  planes  of  projection  is  a  device  for  the  im- 
agination only.  It  explains  the  nature  of  the  "projections,"  or 
"  views,"  which  are  used  in  engineering  drawing. 

5.  Development  or  Flattening  Out  of  the  Planes  of  Projection. — 
Xow  imagine  the  six  sides  of  the  cube  to  be  flattened  out  into  one 
plane  forming  a  grouping  of  six  squares  as  in  Fig.  3.     What  we 


s' 


ir— - 

LI 


Left    Side 
Elevat/on 


H 


"D 

Plan 


^    pROfNT 

Elevation 


y 


BOTTOM 

View 


(Right)  Side 
Elevation 


v 


^    Back 
Elevation 


Fig.  3. 


have  now  is  a  descriptive  or  mechanical  drawing  of  the  object 
showing  six  "  views.''  The  object  itself  is  now  dispensed  with  and 
its  projections  are  used  to  represent  it.  These  six  views  are  what 
we  call  the  "regular  views."  With  one  slight  change  they  cor- 
respond to  the  regular  set  of  drawings  of  a  house  which  architects 
make. 


6 


Engineering  Descriptive  Geometry 


The  set  of  six  "  regular "  projections  would  not  be  altered  by 
passing  the  transparent  planes  at  nnequal  distances  from  each 
other,  so  long  as  they  surround  the  object  and  are  mutually  per- 
pendicular. They  may  form  a  rectangular  parallelopiped  instead 
of  a  cube  without  altering  the  nature  of  the  views. 

It  will  be  noticed  also  that  views  on  opposite  faces  of  the  cube 
differ  but  little.  Corresponding  lines  in  the  interior  may  in  one 
case  be  full  lines  and  in  the  other  "  broken  lines.^'  Broken  lines 
(formed  by  dashes  about  -J"  long,  with  spaces  of  tV")  represent 
parts  concealed  by  nearer  portions  of  the  object  itself.  All  edges 
project  upon  the  plane  faces  of  the  cube,  forming  lines  on  the  draw- 


M 


D 


Plan 


V 


^   Front 
Elevat/on 


s 


^ 


(Ri&HT)  Side- 
Elevation 


H 

Y 

s 

V 

|L 

Plan 

(Right)    Side 
Elevation  - 

X 

V 

0 

eJ 

-RON! 
EVAT 

foN 

Fig.  4. 


Fig.  5. 


ings,  the  edges  concealed  by  nearer  portions  of  the  object  forming 
broken  lines. 

6.  The  Reference  Planes  and  Principal  Views. — In  drawings  of 
parts  of  machinery  six  regular  views  are  usually  unnecessary.  The 
three  views  shown  in  Fig.  2  are  the  "  Principal  Views,"  and  others 
are  needed  only  occasionally.  The  planes  of  those  views  are  the 
"  Eeference  Planes." 

These  views,  when  flattened  from  their  supposed  position  about 
the  object  into  one  plane,  give  the  grouping  in  Fig.  4. 

Another  arrangement  of  the  same  views,  obtained  by  unfolding 
the  planes  of  the  cube  in  a  different  order,  is  shown  in  Fig.  5. 
These  two  arrangements  are  standard  in  mechanical  drawing,  and 
are  those  most  used. 


Nature  of  Orthographic  Projection  7 

7.  The  Nomenclature. — The  nomenclature  adopted  is  as  follows : 
The  "  Eeferenee  Planes,"  or  three  principal  planes  of  projection, 
are  called  from  their  position,  the  Horizontal  Plane,  or  fl,  the 
Vertical  Plane,  or  V,  and  the  (right)  Side  Plane,  or  g.  The  plane 
S  is  by  some  called  the  "  Profile  Plane/'  The  point  0  (Fig.  2), 
in  which  they  meet,  is  the  "  Origin "  of  coordinates.  The  line 
OX,  in  which  H  and  V  intersect,  is  called  the  "  Axis  of  X"  or 
"  Ground  Line.''  The  line  OY,  in  which  ff  and  S  meet,  is  called 
the  "x\xis  of  F/'  and  the  line  OZ,  in  which  V  and  S  meet,  is 
called  the  "  Axis  of  Z."  The  three  axes  together  are  called  the 
"  Axes  of  Projection." 

Since  drawings  are  considered  as  held  vertically  before  the  face, 
it  is  considered  that  the  plane  V  coincides  at  all  times  with  the 
"  Plane  of  the  Paper."  In  unfolding  the  planes  from  their  .posi- 
tions in  Fig.  2  to  that  in  Fig.  4,  it  is  considered  that  the  plane  H 
has  been  revolved  about  the  axis  of  X  (line  OX),  through  an  angle 
of  90°,  until  it  stands  vertically  above  V-  ^^  the  same  way  S  is 
considered  to  be  revolved  about  the  line  OZ,  or  axis  of  Z,  until  it 
takes  its  place  to  the  right  of  V- 

The  arrangement  in  Fig.  5  corresponds  to  a  different  manner  of 
revolving  the  plane  §.  It  is  revolved  about  the  axis  of  Y  (OF) 
until  it  coincides  with  the  plane  H,  and  is  then  revolved  with  H? 
about  the  axis  of  X,  until  both  together  come  into  the  plane  of  the 
paper,  or  V- 

The  three  other  faces  of  the  original  cube  of  planes  of  projection 
are  appropriately  called  H',  V'?  and  S'.  On  account  of  the  simi- 
larity of  the  views  on  them,  to  those  on  H,  V  and  g,  they  are  but 
little  used,  g'  alone  is  fairly  common  since  a  grouping  of  planes 
ff,  V  and  S'  is  at  times  more  convenient  than  the  standard  group 
H,  V  and  S. 

8.  Meaning  of  "  Descriptive  Geometry." — The  aim  of  Engineer- 
ing or  Mechanical  Drawing  is  to  represent  the  shapes  of  solid 
objects  which  form  parts  of  structures  or  machines.  It  shows 
rather  the  shapes  of  the  surfaces  of  the  objects,  surfaces  which  are 
usually  composed  of  plane,  cylindrical,  conical,  and  other  surfaces. 
In  the  drawing  room,  by  the  application  of  mathematical  laws  and 
principles,  views  are  constructed.     These  are  usually  Plan,  Front 


8  -  Engineering  Descriptive  Geometry 

■c 

Elevation,  and  Side  Elevation,  and  are  exactly  such  views  as  would 
be  obtained  if  the  object  itself  were  put  within  a  cage  of  trans- 
parent planes,  and  the  true  projections  formed. 

It  is  these  mathematical  laws  or  rules  which  form  the  subject 
known  as  Descriptive  Geometry.  A  drawing  made  in  such  a  way 
as  to  bring  out  clearly  these  fundamental  laws  of  projection,  by  the 
use  of  axes  of  projection,  etc.,  may  be  conveniently  called  a  "  De- 
scriptive Drawing.'^ 

In  the  practical  application  of  drawing  to  industrial  needs, 
short-cuts,  abbreviations,  and  special  devices  are  much  used  (their 
nature  depending  on  the  special  branch  of  industry  for  which  the 
drawing  is  made).  In  addition,  the  axes  of  projection  are  usually 
omitted  or  left  to  the  imagination,  no  particular  effort  being  made 
to  show  the  exact  mathematical  basis  provided  the  drawing  itself 
is  correct.  Such  a  drawing  is  a  typical  "  Mechanical  Drawing." 
By  the  addition  of  axes  of  projection,  and  similar  devices,  it  may 
be  converted  into  a  strict  "  Descriptive  Drawing." 

9.  The  Descriptive  Drawing  of  a  Point  in  Space. — The  imagi- 
nary process  of  making  a  descriptive  drawing  consists  in  putting 
the  object  within  a  cube  of  transparent  planes,  and  projecting 
points  and  lines  to  these  planes.  In  practice  the  projections  are 
formed  all  on  a  single  sheet  of  paper,  which  is  kept  in  a  perfectly 
flat  shape,  by  the  application  of  rules  of  a  geometrical  kind  de- 
rived from  the  imaginary  process.  The  key  to  the  practical  pro- 
cess is  in  these  rules.  The  first  subject  of  exact  investigation 
should  be  the  manner  of  representing  a  point  in  space  by  its  pro- 
jections and  the  fixing  of  its  position  as  regards  the  "  reference 
planes  "  by  the  use  of  coordinate  distances. 

Figs.  6  and  7  show  the  imaginary  and  the  practical  processes  of 
representing  P  by  its  projections. 

Fig.  6  is  a  perspective  drawing  showing  the  cube  of  planes,  or 
rather  the  three  sides  of  the  cube  regularly  used  for  reference 
planes.  The  cube  must  be  of  such  size  that  the  point  P  falls  well 
within  it.  The  perpendicular  projectors  of  P  are  PPit,  PPv  and 
PPs.  The  origin  and  the  axes  of  projection  are  all  marked  as  on 
Fig.  2. 


Nature  of  Orthographic  Projection 


9 


In  Fig.  7  the  "  field "  of  the  drawing,  that  part  of  the  paper 
devoted  to  it,  is  prepared  by  drawing  two  straight  lines  at  right 
angles  to  represent  the  axes  of  projection,  lettering  the  horizontal 
line  XOYs  and  the  vertical  one  ZOYn.  This  field  corresponds  to 
that  of  Fig.  4,  the  outer  edges  of  the  squares  being  eliminated 
since  there  is  no  need  to  confine  each  plane  to  the  size  of  any  par- 
ticular cube.  If  more  field  is  needed,  the  lines  are  simply  ex- 
tended. It  must  be  remembered  that  these  axes  are  quite  different 
from  the  coordinate  axes  used  in  plane  analytical  geometry,  or 
graphic  algebra.     These  divide  the  field  of  the  drawing  into  four 


i  H 


X~1e  ^  o' 


Pv        ^         9 
z 


A 


—  J 

p. 


Fig.  6. 


Fig.  7. 


quadrants,  of  which  three  represent  three  different  planes,  mutu- 
ally perpendicular,  the  fourth  being  useful  only  for  the  purposes 
of  construction. 

Usually  the  upper  left  quadrant,  the  "  North-West,"  represents 
IHI ;  the  lower  left  quadrant,  or  "  South-West,"  represents  V>  and 
the  lower  right  quadrant,  or  "  South-East,"  represents  g. 

On  occasion  the  axes  may  be  lettered  XOZs  horizontally  and 
ZvOY  vertically,  to  correspond  to  Fig.  5,  the  upper  right  quadrant 
now  representing  §. 

10.  Coordinates  of  a  Point  in  Space. — A  point  in  space  is 
located  by  its  perpendicular  distances  from  the  three  planes  of 
projection,  that  is  to  say,  by  the  length  of  its  projectors.     These 


10  Engineering  Descriptive  Geometry 

distances  are  called  the  coordinates  of  the  point,  and  are  designated 
by  X,  y  and  z.  In  the  example  given,  these  values  are  2,  3  and  1. 
In  Fig.  6  PPs,  the  S  projector  of  P,  is  two  units  long,  or  x  —  %. 
The  perpendicular  distance  to  the  plane  V^  "the  V  projector,  PPv, 
is  three  units  long.  y=^.  In  the  same  way  PPn,  the  H  projector, 
is  one  unit  long.    z  =  l. 

In  describing  the  point  P,  it  is  sufficient  to  state  that  it  is  the 
point  for  which  a;  =2,  y='^y  and  2  =  1.  This  is  abbreviated  con- 
veniently by  calling  it  the  point  P  (2,  3, 1),  the  coordinates,  given 
in  the  bracket,  being  taken  always  in  the  order  x,  y,  z. 

The  projectors,  the  true  coordinate  distances,  are  shown  in  Fig. 
6  by  lines  of  dots,  not  dashes. 

If  in  each  plane  H,  V  and  §,  perpendicular  lines  are  drawn 
(dashes,  not  dots)  from  the  projections  of  P  to  the  axes,  we  shall 
have  the  lines  PhS  and  Pa/,  P^e  and  Px-g,  Psg  and  Psf.  These  lines 
meet  in  pairs  at  e,  g,  and  f,  forming  a  complete  rectangular  paral- 
lelopiped  of  which  P  and  0  are  the  extremities  of  a  diagonal.  The 
other  corners  of  the  parallelopiped  are  Ph,  Pv,  Ps,  e,  f  and  g. 

Each  coordinate,  x,  y  and  z,  appears  in  four  places  along  four 
edges  of  the  parallelopiped,  as  is  marked  in  Fig.  6. 

The  distances  x,  y  and  z  are  all  considered  positive  in  the  case 
shown. 

In  Fig.  7,  the  descriptive  drawing  of  the  point  P,  P  itself  does 
not  appear,  being  represented  by  its  projections,  Pn,  Pv  and  P,. 
The  true  projectors  (shown  in  Fig.  6  by  lines  of  dots)  do  not 
appear,  but  in  place  of  each  coordinate  tliree  distances  equal  to  it 
do  appear,  so  that  in  Fig.  1  x,  y  and  z  each  appear  in  three  places 
as  is  there  marked.  Thus  x  appears  as  Phfn,  eO,  and  P^g.  As  all 
these  are  measured  to  the  left  from  the  vertical  axis,  ZOYn,  it 
follows  that  Phepv  is  a  straight  line,  or  Pn  is  vertically  above  Pv 
It  is  often  said  that  Pv  "  projects  vertically  "  to  Ph.  In  the  same 
way  Pv  ^'  projects  horizontally  "  to  Pg.  The  distance  y  appears  as 
ePji,  Ofhf  Ofs,  and  gPg.  The  point  /  appears  double  due  to  the 
axis  of  Y  itself  doubling.  To  represent  the  original  coincidence  of 
fh  and  fs,  a  quadrant  of  a  circle  with  center  at  0  is  often  used  to 
connect  them. 


Nature  of  Orthographic  Projection  11 

11.  Three  Laws  of  Projection  for  H,  V  and  S-— The  three  rela- 
tions shown  by  Fig.  7  amount  to  three  laws  governing  the  pro- 
jections of  a  point  in  the  three  views,  and  must  always  be  rigidly 
observed.  They  may  seem  easy  and  obvious  when  applied  to  one 
point,  but  when  dealing  with  a  multitude  of  points  it  is  not  easy 
to  observe  them  unfailingly. 

They  may  be  thus  tabulated : 

(1)  Ph  must  be  vertically  above  Pi,. 

(2)  Pg  must  be  on  the  same  horizontal  line  as  Pv. 

(3)  Ps  must  be  as  far  to  the  right  of  OZ  as  Pn  is  above  OX. 
From  these  laws  it  follows  that  if  two  projections  of  a  point  are 

given,  the  third  is  easily  found.  In  Fig.  7,  if  two  of  the  corners 
of  the  figure  PhfhfsPsPv  are  given,  the  figure  can  be  graphically 
completed.  Much  of  the  work  of  actual  mechanical  drawing  con- 
sists in  correctly  locating  two  of  the  projections  of  a  point  by  plot- 
ting or  measuring,  and  of  finding  the  other  projection  by  the  appli- 
cation of  these  laws  or  of  this  construction.  Constant  checking 
of  the  points  between  the  various  views  of  a  drawing  is  a  vital  prin- 
ciple in  drawing. 

On  the  drawing  board  the  horizontal  projection  of  Pv  to  P«  is 
naturally  done  by  the  T-square  alone,  and  the  vertical  projection 
of  Ph  to  Pv  by  T-square  and  triangle.  There  are  two  methods  of 
carrying  out  the  third  law  in  addition  to  the  graphical  construc- 
tion of  Fig.  7.  Fig.  8  shows  a  graphical  method  which  makes  use. 
of  a  45°  line,  OL,  in  the  construction  space,  instead  of  the  quad-- 
rant  of  a  circle.  It  is  easier  to  execute,  but  the  meaning  is  not  so 
clearly  shown.  The  third  method  is  by  the  use  of  the  dividers, 
directly  to  transfer  the  x  coordinate  from  whichever  place  it  is. 
fi^st  plotted,  to  the  other  view  in  which  it  appears. 

12.  Paper  Box  Diagrams. — When  studying  a  descriptive  draw- 
ing, such  as  Fig.  8,  imagine  as  you  look  at  Pv  that  the  real  point  P 
lies  haclc  of  the  paper,  at  a  distance  equal  to  ePn. 

Whenever  figures  in  the  text  following  seem  hard  to  grasp,  carry 
out  the  following  scheme.  Trace  the  figure  on  thin  paper,  or  on 
tracing  cloth.  Using  Fig.  8  as  an  example,  and  supposing  it  to 
have  been  traced  ^on  semitransparent  paper,  hold  the  paper  before 
you  and  fold  the  top  half  back  90°  on  the  line  XOYs.    Then,  view- 


12 


Engineering  Descriptive  Geometry 


ing  Ph  from  above,  imagine  the  true  point  P  to  lie  below  the  paper 
at  a  distance  equal  to  ePv,  in  the  same  way  as  you  imagine  P  to 
lie  back  of  Pv  at  a  distance  equal  to  ePh, 

After  flattening  the  paper,  fold  the  right  half  back  90°  on  the 
line  ZOYh,  and,  viewing  Ps  from  the  right,  imagine  P  to  lie  back 
of  Ps  a  distance  Pvff.  Finally,  crease  the  paper  on  the  line  OL, 
OL  itself  forming  a  groove,  not  a  ridge,  and  bend  the  paper  on  all 


^ 

A 

1  IH 
i 

1 
j 

"\ 

/ 

X   jC 

0 

1        '      4 

Y, 

1     ..    ... 

Pv 

V 

~'9 

_..    ..    „    ..    J 
P3 

z 

• 

FiQ.  8. 


the  creases  at  once,  so  that  H  and  S  fold  back  into  positions  at 
right  angles  to  V  and  to  each  other  at  the  same  time. 

The  "  construction  space  '^  YhOYs  is  thus  folded  away  inside 
and  OYh  and  OYs  come  in  contact  with  each  other.  Fig.  9  shows 
the  final  folding  partly  completed. 

'No  diagram,  however  complicated,  can  remain  obscure  if  studied 
from  all  sides  in  this  manner. 

To  have  a  convenient  name,  these  space  diagrams  may  be  called 
"  Paper  Box  Diagrams." 


Nature  of  Orthographic  Projection 


13 


Figs.  4  and  5  make  good  paper  box  diagrams,  while  Fig.  3  may- 
be traced  and  folded  into  a  perfect  cube  which,  if  held  in  proper 
position,  will  give  the  exact  views  shown  in  Figs.  2  and  2a,  omit- 
ting the  solid  object  supposed  to  be  seen  in  the  center  of  those 
figures. 

13.  Zero  Coordinates. — Points  having  zero  coordinates  are  some- 
times perplexing.  If  one  coordinate  is  zero,  the  point  in  question 
is  on  one  of  the  reference  planes,  and  indeed  coincides  with  one  of 
its  own  projections.  Since  x  is  the  length  of  the  orthographic 
projector  of  the  point  P  upon  the  plane  S>  if  a:=0,  this  projector 
disappears  and  the  point  P  and  its  S  projection  Ps  coincide.     If 


Fig.  9. 


the  point  Q  (0,  3, 1)  is  to  be  plotted  it  will  be  found  to  coincide 
with  Ps  in  Fig.  6.  The  descriptive  drawing  will  correspond  with 
Fig.  7  with  all  lines  to  the  left  of  ZOYn  omitted,  and  with  the  let- 
tering changed  as  follows:  For  Ps  put  Qs  (and  Q),  for  fu  put  Qn, 
for  g  put  Qv  JThe  student  should  make  this  diagram  on  cross- 
section  paper  and  should  study  out  for  himself  the  similar  cases  for 
the  points  Q'  (2,  0, 1)  [P^  in  Fig.  6]  and  Q"  (2,  3,  0)  [Pn  in  Fig. 
6]  and  should  proceed  from  them  to  more  general  cases,  assuming 
ordinates  at  will,  using  cross-section  paper  for  rapid  sketch  work 
of  this  kind. 

If  two  coordinates  are  zero,  the  point  lies  on  one  of  the  axes, 
on  that  axis,  in  fact,  which  corresponds  to  the  ordinate  which  is  not 
zero.  Thus  the  point  R  (2,  0,  0)  is  the  point  e  of  Fig.  6,  En  and 
Bv  are  at  e,  and  Rg  is  at  0. 


Fig.  10. 


Ftg.  10a. 

These  wire-mesh  cages  are  not  essential  for  a  clear  understanding 
of  the  course.  Cross-section  paper  should  be  used  in  the  solution 
of  the  problems  and  folded  to  make  "  space  "  or  "  paper  box  "  dia- 
grams, to  illustrate  knotty  points.  These  folded  diagrams  are 
practically  miniature  cages.  The  full-size  cages  are  very  con- 
venient for  class-room  demonstrations. 


16  Engineering  Descriptive  Geometry 

Wire-mesh  Cage. 

If  possible,  it  is  very  desirable  to  have  cages  similar  to  Fig.  10," 
formed  of  wire-mesh  screens,  representing  the  planes  ff,  V?  S  and 
S'.  On  these  screens  chalk  marks  may  be  made  and  the  planes, 
being  hinged  together,  may  afterward  be  brought  into  coincidence 
with  Vj  as  represented  in  Fig.  10a. 

In  order  to  plot  points  in  space  within  the  cage,  pieces  of  wire 
about  20  inches  long,  with  heads  formed  in  the  shape  of  small 
loops  or  eyes,  are  used  as  point  markers.  They  may  be  set  in  holes 
drilled  in  the  base  of  the  cage  at  even  spaces  of  1"  in  each  direc- 
tion, so  that  a  marker  may  be  set  to  represent  any  point  whose  x 
or  y  coordinates  are  even  inches.  To  adjust  the  marker  to  a  re- 
quired z  coordinate,  it  may  be  pulled  down  so  that  the  wire  projects 
through  the  base,  lowering  the  head  the  required  amount,  z  may 
vary  fractionally. 

In  Fig.  10  a  point  marker  is  set  to  the  point  P  (11,  4,  6),  and 
the  lines  on  the  screens  have  been  put  on  with  chalk,  to  represent 
all  the  lines  analogous  to  those  of  Fig.  6. 

Fig.  10a  represents  the  descriptive  drawing  produced  by  the 
development  of  the  screens  in  Fig.  10.     It  is  analogous  to  Fig.  7. 

Several  points  may  be  thus  marked  in  space  and  soft  lead  wire 
threaded  through  the  loops,  so  that  any  plane  figure  may  be  shown 
in  space,  and  its  corresponding  orthographic  projections  may  be 
drawn  on  the  planes  in  chalk. 

Problems  I. 

(For  solution  with  wire-mesh  cage.) 

1.  Plot  by  the  use  of  the  wire  markers  the  three  points,  A,  B 
and  C,  whose  coordinates  are  (5,12,11),  (3,3,3),  and  (12,4,8), 
and  draw  the  projections  on  the  screens  in  chalk.  By  joining  point 
to  point  a  triangle  and  its  projections  are  formed.  Use  lead  wire 
for  joining  the  points,  and  chalk  lines  for  joining  the  projections. 

2.  Form  the  triangle  as  above  with  the  following  coordinates: 

(11,  3,  2),  (12,  6, 12)  and  (14, 12,  7). 

3.  Form  the  triangle  as  above  with  the  following  coordinates: 

(7,0,11),   (9,9,0)   and   (2,2,3). 


Nature  of  Orthographic  Projection  17 

4.  Form  the  triangle  as  above  with  the  following  coordinates: 

(0,11,13),  (14,3,3)  and  (14,13,0). 
(The  following  examples  may  be  solved  on  coordinate  paper,  or 
plotted  in  inches  on  the  blackboard.) 

5.  Make  the  descriptive  drawing  of  a  triangle  in  three  views  by 
plotting  the  vertices  and  joining  them  by  straight  lines.  The 
vertices  are  the  points  A  (1, 10,  8),  B  (5,  6,  8),  C  (9,  2,  4). 

6.  Make  the  descriptive  drawing  as  above  using  the  points 

A  (12,  2,  5),  ^  (0,  8,  6),  (7  (4,  6,  0). 

7.  Make  the  descriptive  drawing  as  above  using  the  points 

A   (3,4,2),  B  (13,8,10),  C  (5,10,14). 

8.  The  four  points  A  (3,  3,  3),  B  (3,  3, 15),  C  (15,  3, 15),  and 
D  (15,3,3)  form  a  square.  Make  the  descriptive  drawing.  Why 
are  two  projections  straight  lines  only?  What  are  the  coordinates 
of  the  center  of  the  square  ? 

9.  The  four  points  A  (12,2,12),  B  (2,2,12),  C  (7,14,12), 
and  D  (7,  6,  2)  are  the  comers  of  a  solid  tetrahedron.  Make  the 
descriptive  drawing,  being  careful  to  mark  concealed  edges  by 
broken  lines. 

10.  Make  the  descriptive  drawing  of  the  tetrahedron  A  (2,  3,  2), 
B  (9,  8,  3),  (7  (4,  8,  9),  i)  (12,3,6),  marking  concealed  edges  by 
broken  lines. 

11.  Make  the  descriptive  drawing  of  the  tetrahedron  A  (3,  2,  4), 
B  (6,8,2),  C  (8,1,8),I>  (2,7,8). 

12.  Plot  the  points  A  (12,7,7),  B  (8,13,5),  C  (2,9,2),  and 
D  (6,3,4).    Why  is  the  V  projection  a  straight  line? 

13.  Make  the  descriptive  drawing  of  the  tetrahedron  A  (13,  5,  3), 
B  (1,  5,  3),  C  (7,  2,  6),  D  (7,  8,  6).  To  which  axis  is  the  line  AB 
parallel  ?    To  which  axis  is  CD  parallel  ? 

14.  Plot  and  join  the  points  4  (11,  3,  3),  5  (3,  3,  3),  (7  (7,  9,  7), 
and  Z>  (15,  9,  7).  Bo  AC  and  BD  meet  at  a  point  or  do  they  pass 
without  meeting? 


CHAPTER  II. 

ORTHOGRAPHIC   PROJECTION   OF   THE   FINITE   STRAIGHT 

LINE. 

14.  The  Finite  Straight  Line  in  Space :  One  not  Parallel  to  any 
Reference  Plane,  or  an  "  Oblique  Line." — A  line  of  any  kind  con- 
sists merely  of  a  succession  of  points.  Its  orthographic  projection 
is  the  line  formed  by  the  projections  of  these  points. 

In  the  case  of  a  straight  line,  the  orthographic  •  projection  is 
itself  a  straight  line,  though  in  some  cases  this  straight  line  may 
degenerate  to  a  single  point,  as  mathematicians  express  it. 


Fig.  11. 


FiQ.  12. 


To  find  the  fi-jl,  V  and  §  projections  of  a  finite  straight  line  in 
space,  the  natural  course  is  to  project  the  extremities  of  the  line 
on  each  reference  plane  and  to  connect  the  projections  of  the  ex- 
tremities by  straight  lines.  We  shall  not  consider  this  as  requir- 
ing proof  here.  It  is  common  knowledge  that  a  straight  line  cannot 
be  held  in  any  position  that  will  make  it  appear  curved,  and  ortho- 
graphic projection  is,  as  shown  by  Fig.  1,  only  a  special  case  of 
perspective  projection.  The  strict  mathematical  proof  is  not  ex- 
actly a  part  of  this  subject. 


Orthographic  Projectiox  of  Finite  Straight  Line     19 


The  projectors  from  the  different  points  of  a  straight  line  form 
a  plane  perpendicular  to  the  plane  of  projection.  This  "  projector- 
plane,"  of  course,  contains  the  given  line.  If  the  straight  line  is 
a  limited  or  finite  line  the  projector-plane,  is  in  the  form  of  a 
quadrilateral  having  two  right  angles.  Thus  in  Fig.  11  the  M 
projectors  of  the  straight  line  AB  form  the  figure  AAnBhB,  having 
right  angles  at  Ah  and  Bn.  These  projector-planes  AAhBnB, 
AAsBsB,  and  AAvBvB  are  shown  clearly  in  this  perspective  draw- 
ing, in  which  they  are  shaded. 

Fig.  12  is  the  descriptive  drawing  of  the  same  line  AB  which  has 
been  selected  as  a  "line  in  space,"  that  is,  as  one  which  does  not 
obey  any  special  condition.  In  such  general  cases  the  projections 
are  all  shorter  than  the  line  itself.  As  drawn,  the  extremities  are 
A  (1,1,5)  and  B  (5,4,2). 


-I — I — I — h 


FiG.  13. 


Fig.  14. 


1 1 r        t 


T. 


As  B. 


15.  Line  Parallel  to  One  Reference  Plane,  or  Inclined  Line. — 

A  line  which  is  parallel  to  one  reference  plane,  but  is  not  parallel 
to  an  axis,  appears  projected  at  its  true  length  on  that  reference 
plane  only. 

Figs.  13  and  14  show  a  line  five  units  long,  connecting  the  points 
A  (1,2,2)  and  B  (5,5,2).  AnBn  is  also  five  units  in  length  but 
AvBr:  is  but  four  and  AgBg  is  three.  The  projector-plane  AAhBhB 
is  a  rectangle. 

The  student  should  construct  on  coordinate  paper  the  two  simi- 
lar cases.  For  example:  the  line  C  (4,  2, 1),  Z)  (1,  2,  5)  is  parallel 
to  V;  ^  (2,1,2),  F  (2,5,5)  is  parallel  tog. 


20 


Engineering  Descriptive  Geometry 


16.  Line  Parallel  to  One  of  the  Axes  and  thus  Parallel  to 
Two  Reference  Planes. — If  a  finite  straight  line  is  parallel  to  one 
of  the  axes  of  projection,  its  projection  on  the  two  reference  planes 
which  intersect  at  that  axis,  will  be  equal  in  length  to  the  line 
itself.  Its  projection  on  the  other  reference  plane  will  be  a  single 
point. 

Fig.  15  is  the  perspective  drawing  and  Fig.  16  the  descriptive 
drawing,  of  a  line  parallel  to  the  axis  of  X,  four  units  in  length. 
Its  extremities  are  the  points  A  (1,2,2)  and  5  (5,  2,  2).     In  H 


B. 


e 


V 


% . 


H — I — I — 1-— I — h 


FiG.  15. 


Av 

Z 

Fig.  16. 


V. 


5 


and  V  its  projections  are  four  units  long.  The  projector-planes 
AAjiBhB  and  AAvBvB  are  rectangles.  The  §  projector-plane  de- 
generates to  a  single  line  BAAg.  It  will  be  seen  that  the  coordi- 
nates of  the  extreme  points  of  the  line  differ  only  in  the  value  of 
the  X  coordinate.  In  fact,  any  point  on  the  line  will  have  the 
y  and  z  coordinates  unchanged.    It  is  the  line  (x  variable,  2,  2). 

The  student  should  construct  for  himself  descriptive  drawings 
of  lines  parallel  to  the  axis  of  Y  and  the  axis  of  Z,  using  prefer- 
ably "  coordinate  paper "  for  ease  of  execution.  Good  examples 
are  the  lines  C  (1, 1, 1),  P  (1,5,1)  and  JS'  (3,1,1),  F  (3,1,4). 
Points  on  the  line  CD  differ  only  as  regards  the  y  coordinate.  It 
is  a  line  parallel  to  the  axis  of  Y.  EF  is  parallel  to  the  axis  of  Z 
and  z  alone  varies  for  different  points  along- the  line. 


Orthographic  Projection  of  Finite  Straight  Line     21 

17.  Foreshortening. — The  projection  of  a  line  oblique  to  the 
plane  of  projection  is  shorter  than  the  original  line.  This  is 
called  foreshortening.  The  H>  V  and  S  projections  of  Fig.  12, 
and  the  V  and  S  projections  of  Fig.  14,  are  foreshortened.  It  is 
a  loose  method  of  expression,  but  a  common  one,  to  say  that  a  line 
is  foreshortened  when  it  is  meant  that  a  certain  projection  of  a 
line  is  shorter  than  the  line  itself.  When  subscripts  are  omitted 
and  AhBh  is  called  AB,  it  is  natural  to  speak  of  the  line  ^4^  as 
appearing  "foreshortened"  in  tlie  plan  view  or  projection  on  ff. 
This  inexact  method  of  expression  is  so  customary  that  it  can 
hardly  be  avoided,  but  with  this  explanation  no  misconception 
should  be  possible. 

18.  Inclined  and  Oblique  Lines. — The  words  Inclined  and  Obli- 
que are  taken  generally  to  mean  the  same  thing,  but  in  this  subject 
it  becomes  necessary  to  draw  a  distinction,  in  order  to  be  able  to 
specify  without  chance  of  misunderstanding  the  exact  nature  of  a 
given  line  or  plane. 

A  line  will  be  described  as : 
Parallel  to  an  axis,  when  parallel  to  any  axis.    As  a  special  case  a 

line  parallel  to  the  axis  of  Z  may  be  called  simply  vertical. 
Inclined,  when  parallel  to  a  reference  plane,  but  not  parallel  to  an 

axis.    The  line  AB,  Fig.  13,  is  an  illustration. 
Oblique,  when  not  parallel  to  any  reference  plane  or  axis.     The 

typical  "  line  in  space"  is  oblique.    AB,  of  Fig.  11,  illustrates 

this  case. 

19.  Inclined  and  Oblique  Planes. — A  plane  will  be  called: 
Horizontal,  when  parallel  to  ff.    The  V  projector-plane  in  Fig.  15 

is  of  this  kind. 
Vertical,  when  parallel  to  V  o^  S*    The  H  projector-plane  in  Fig. 

15  is  of  this  kind. 
Inclined,  when  perpendicular  to  one  reference  plane  only.    The  H 

projector-plane  of  Fig.  13  is  of  this  kind. 
Oblique,  when  not  perpendicular  to  any  reference  plane.     Planes 

of  this  kind  will  appear  later  on. 
The  surface  of  the  solid  of  Fig.  2  is  composed  of  vertical,  hori- 
zontal, and  inclined  planes  (but  no  oblique  plane).     Its  edges  are 


22 


Engineerixg  Descriptive  Geometry 


lines,  parallel  to  the  axes  of  X,  Y  and  Z;  and  inclined  lines  (be- 
cause parallel  to  §)  ;  but  no  oblique  lines. 

20.  The  Point  on  a  Given  Line. — It  is  self-evident  that  if  a 
given  point  is  on  a  given  line,  all  the  projections  of  the  point  must 
lie  on  the  projections  of  the  line. 

If  the  middle  point  of  a  line  AB  is  projected,  as  C  in  Fig.  17,  its 
projections  Cj,,  Cv,  and  Ca  bisect  the  projections  of  the  line.  The 
reason  for  this  appears  when  we  consider  the  true  shape  of  the 


Fig.  17. 


projector-planes,  all  three  of  which  appear  distorted  in  the  per- 
spective drawing.  Fig.  17,  and  which  do  not  appear  at  all  on  the 
descriptive  drawing.  Fig.  18.  In  Fig.  17  AAhBhB  is  a  quadri- 
lateral, having  right  angles  at  ^^  and  Bh,  it  is  therefore  a  trape- 
zoid. CCh  is  parallel  to  AAn  and  BBh,  and  since  it  bisects  AB  at  C, 
it  must  also  bisect  AhBh  at  Ch.  The  result  of  this  is  that  in  Fig. 
18,  where  the  projections  which  do  appear  are  of  their  true  size, 
Ch  bisects  AjiBj,,  Co  bisects  AvB^,  and  Cs  bisects  AgBg. 

This  principle  applies  to  other  points  than  the  bisector.  Since 
all  W  projectors  are  parallel  to  each  other,  if  any  point  divides  AB 
into  unequal  parts,  the  projections  of  the  point  will  divide  the 
projectors  of  AB  in  parts  having  the  same  ratio.     A  point  one- 


Orthographic  Projection  of  Finite  Straight  Line     23 

tenth  of  the  distance  from  A  to  5  will,  by  its  projections,  mark 
off  one-tenth  of  the  distance  on  AhBn,  AvBv,  etc. 

The  points  illustrated  in  Figs.  17  and  18  are  A  (2,3,1), 
^  (5,  5,  5)  and  C  (3J,  4,  3).  It  will  be  noticed  that  the  x  coordi- 
nate of  C  is  the  mean  of  those  of  A  and  B,  and  the  y  and  z  coordi- 
nates of  C  also  are  the  mean  of  the  y  and  z  coordinates  of  A  and  B. 

Unless  all  three  of  the  projections  of  a  point  fall  on  the  pro- 
jections of  a  line,  the  point  is  not  in  the  given  line.  If  one  of  the 
projections  of  the  point  be  on  the  corresponding  projection  of  the 
line,  one  other  projection  of  both  point  and  line  should  be  ex- 
amined. If  in  this  second  projection  it  is  found  that  the  point 
does  not  lie  on  the  line,  it  shows  that  the  point  in  space  lies  in  one 
of  the  projector-planes. 

Thus  the  point  D  in  Fig.  18  has  its  V  projection  on  AvBv,  but 
its  fi  and  S  projections  are  not  on  AhBh  and  AgBs.  D  is  not  a 
point  in  the  line  AB  but  is  on  the  V  projector-plane  of  AB,  as  is 
clearly  shown  on  Fig.  17. 

In  the  case  illustrated,  Dv  bisects  BvCv.  The  plotting  of  the  V 
projection  of  a  point  is  governed  only  by  its  x  and  z  coordinates. 
Dv  bisects  BvC'v  because  its  x  and  z  coordinates  are  the  means  of 
the  X  and  z  coordinates  of  B  and  C.  The  y  coordinate  of  D,  how- 
ever, has  no  connection  with  the  y  coordinates  of  B  and  C. 

21.  The  Isometric  Diagram. — A  device  to  obtain  some  of  the 
realistic  appearance  of  a  true  perspective  drawing  without  the 
excessive  labor  of  its  construction  is  known  as  "  isometric ''  draw- 
ing. 

A  full  explanation  of  this  kind  of  drawing  will  follow  later, 
but  for  present  purposes  we  may  regard  it  as  a  simplified  per- 
spective of  a  cube  in  about  the  position  of  that  in  Figs.  2,  6,  11, 
etc.,  but  turned  a  little  more  to  the  left.  Vertical  lines  are  un- 
changed. Lines  which  are  parallel  to  the  axis  of  X,  and  which  in 
the  perspective  drawing  incline  up  to  the  left  at  various  angles,  are 
all  made  parallel  and  incline  at  30°  to  the  horizontal.  In  the- 
same  way  lines  parallel  to  the  axis  of  Y  are  drawn  at  30°  to  the 
horizontal,  inclining  up  to  the  right. 


24 


Engineering  Descriptive  Geometry 


Fig.  19  shows  the  shape  of  a  large  cube  divided  into  small  unit 
cubes.  In  plotting  points  the  same  scale  is  used  in  all  three  direc- 
tions, that  is,  for  distances  parallel  to  all  three  axes.  Fig.  19a 
shows  the  point  P  (2,  3, 1)  plotted  in  this  manner,  so  that  the 
figure  is  equivalent  to  the  true  perspective  drawing.  Fig.  6. 

It  is  not  intended  that  the  student  should  make  any  true  per- 
spective drawing  while  studying  or  reciting  from  this  book.  If  any 
of  the  space  diagrams  here  shown  by  true  perspective  drawings 


Fig.  19. 


Fig.  19a. 


must  be  reproduced,  the  corresponding  isometric  drawing  should 
be  substituted. 

For  rapid  sketch  work,  especially  ruled  paper,  called  "  isometric 
paper,'^  is  very  convenient.  It  has  lines  parallel  to  each  of  the 
three  axes.  With  such  paper  it  is  easy  to  pick  out  lines  correspond- 
ing to  those  of  Fig.  19. 

An  excellent  exercise  of  this  kind  is  to  sketch  on  isometric 
paper  the  shaded  solid  shown  in  Fig.  2,  taking  the  unit  square  of 
the  paper  for  1"  and  considering  the  solid  to  be  cut  from  a  10"  cube, 
the  thickness  of  the  walls  left  being  2",  and  the  height  of  the  tri- 
angular portion  being  6".  The'  solid  may  be  sketched  in  several 
positions. 


Orthographic  Projection  of  Finite  Straight  Line     25 

Problems  II. 

(For  solution  with  wire-mesh  cage,  or  cross-section  paper,  or  on 

blackboard.) 

15.  A  line  connects  the  points  A  (5|^,  6)  and  B  (5, 12,  G). 
What  are  the  coordinates  of  the  point  C,  the  center  of- the  line? 
What  are  the  coordinates  of  D,  a  point  on  the  line,  one- tenth  of 
the  way  from  A  to  Bf 

16.  Same  with  points  A  (6,  6,  2)  and  5  (6,  6, 12). 

17.  Draw  the  line  AB  whose  extremities  are  A  (2,  7,  4)  and 
B  (14,  2,  4).  On  what  view  does  its  true  length  appear?  What  is 
this  length?  What  are  the  coordinates  of  a  point  C  on  the  line 
one-third  of  its  length  from  A  ? 

18.  With  the  same  line  A  (2,7,4),  B  (14,2,4),  state  what  is 
the  true  shape  of  the  H  projector-plane.  Give  length  of  each  edge 
and  state  what  angles  are  right  angles.  Same  for  V  projector- 
plane. 

19.  Same  as  Problem  18,  with  line  A  (4,  2,  2),  B  (4, 11,  8). 

20.  With  the  line  of  Problem  19,  state  what  is  the  true  shape  of 
the  H  and  S  projector-planes,  giving  length  of  each  edge,  and 
state  which  angles  are  right  angles. 

21.  Same  as  Problem  17,  with  line  A  (0,  4,  8),  5  (9,  4, 1). 

22.  The  H  projection  of  C  (8,2,  6)  lies  on  the  fil  projection  of 
the  line  A  (10, 1,  9),  B  (2,  5,  2).    Is  the  point  on  the  line?     . 

23.  Same  as  Problem  22,  with  line  A  (2, 1,  8),  B  (8, 10,  5),  and 
point  C  (4,4,7). 

24.  A  triangle  is  formed  by  joining  the  points  A  (6,  3, 1), 
B  (10,3,10)  and  C  (2,3,10).  In  what  view  or  views  does  the 
true  length  oi  AB  appear?  In  what  view  or  views  does  the  true 
length  of  BC  appear?  Mark  the  center  of  the  triangle  (one-third 
the  distance  from  the  center  of  the  base  BC  to  the  vertex  A)  and 
give  its  coordinates. 

25.  Same  with  points  A  (5,9,6),  B  (5,3,1)  and  C  (5,3,11). 

26.  Same  with  points  A  (10, 1,  4),  5  (7, 10,  4)  and  C  (1,  4,  4). 

27.  The  V  projections  of  the  points  A  (8,  1,  2),  B  (10,  3,  8), 
C  (4,  3, 10.)  and  D  (2, 1,  4)  form  a  square.  Draw  the  projections 
and  connect  them  point  to  point.  What  are  the  coordinates  of  the 
center  where  AC  and  BD  intersect? 


26  Engineering  Descriptive  Geometry 

28.  Plot  the  parallelogram  A  (11,  3,  3),  B  (3,  3,  3),  C  (7,  9,  7), 
D  (15,  9,  7).  The  diagonals  intersect  at  E.  Give  the  coordinates 
of  E.  Describe  the  ff  projector-planes  of  AB,  AC,  and  CD,  giving 
the  length  of  each  end  projector.  Is  the  plane  of  the  figure  in- 
clined or  oblique?    Is  AC  an  inclined  or  an  oblique  line? 

29.  Plot  the  quadrilateral 

A   (11, 10,3),  5  (3,10, 11),  0  (7,2,  7),  D  (11,4,  3). 

Is  the  plane  of  the  figure  horizontal,  vertical,  inclined  or  oblique? 

Is  the  line  AB  horizontal,  vertical,  inclined  or  oblique  ? 

Is  the  line  BC  horizontal,  vertical,  inclined  or  oblique? 

Is  the  line  CD  horizontal,  vertical,  inclined  or  oblique? 

Is  the  line  DA  horizontal,  vertical,  inclined  or  oblique? 


CHAPTER  III. 
THE  TETTE  LENGTH  OF  A  LINE  IN  SPACE. 

22.  The  Use  of  an  Auxiliary  Plane  of  Projection. — To  find  the 
true  length  of  a  "  line  in  space/'  or  oblique  straight  line,  an  auxil- 
iary plane  of  projection  is  of  great  value,  and  is  constantly  used 
in  all  branches  of  Engineering  Drawing. 

A  typical  solution  is  shown  by  Figs.  20  and  21.  The  essential 
feature  is  the  selection  of  a  new  plane  of  projection,  called  an 


Fig.  20. 


Fig.  21. 


auxiliary  plane,  and  denoted  by  \J,  which  must  be  parallel  to  the 
given  line  and  easily  revolved  into  coincidence  with  one  of  the 
regular  planes  of  reference. 

This  auxiliary  plane  is  passed  parallel  to  one  of  the  projector- 
planes.  In  Fig.  20  the  plane  S'  of  the  cube  of  planes  has  been 
replaced  by  a  plane  \],  parallel  to  the  ff  projector-plane,  AAhBnB. 
Like  that  plane,  \J  is  also  perpendicular  to  fi,  and  XM,  the  line 
of  intersection  of  U  and  H,  is  parallel  to  AhBn.  The  distance  of 
the  plane  U  from  the  projector-plane  may  be  taken  at  will  and 
in  the  practical  work  of  drawing  it  is  a  matter  of  convenience, 
choice  being  governed  by  the  desire  to  make  the  resulting  figures 
clear  and  separated  from  each  other.     In  Fig.  20  the  auxiliary 


28  Engineerixg  Descriptive  Geometry 

plane  U  has  been  established  by  selecting  a  point  X  in  H  for  it 
to  pass  through.    U  is  an  "  inclined  plane,"  not  an  "  oblique  plane/' 

23.  Traces  of  the  Auxiliary  Plane  \]. — ^The  auxiliary  piano  U 
cuts  the  plane  V  iii  ^  ^i^e  XN,  parallel  to  the  axis  of  Z.  The 
lines  of  intersection  of  \]  with  the  reference  planes,  are  called  the 
"traces'^  of  \].  Since  there  are  three  reference  planes,  there  may 
be  as  many  as  three  traces  of  UJ.  In  the  case  illustrated  in  Fig.  20, 
there  are,  however,  but  two  traces.  Qnly  one  of  these  three'  possible 
traces  of  UJ  can  be  an  inclined  line.  In  Fig.  20  the  trace  XM  alone 
is  an  "  inclined  "  line. 

We  shall  see  later  that  the  auxiliary  plane  may  be  taken  per- 
pendicular to  V  or  to  §  as  alternative  methods.  In  every  case 
there  is  but  one  inclined  trace,  that  on  the  plane  to  which  U  is 
perpendicular.  It  is  this  trace  which  lias  the  greatest  importance  in 
the  process.  For  the  sake  of  uniformitj^,  M  and  N  will  be  assigned 
as  the  symbols  for  marking  the  traces  of  an  auxiliary  plane  of 
projection. 

24.  The  U  Projectors. — A  new  system  of  projectors,  AAu,  BBu, 
etc.,  project  the  line  AB  upon  the  plane  U-  These  projectors, 
being  perpendicular  distances  between  a  line  and  a  plane  parallel  to 
it,  are  all  equal,  and  the  projector-plane  AAuBuB  of  Fig.  20  is  in 
reality  a  rectangle.  AuBu  is  therefore  equal  in  length  to  AB,  or 
AB  is  projected  upon  U  without  foreshortening. 

25.  Development  of  the  Auxiliary  Plane  JJ- — The  descriptive 
drawing.  Fig.  21,  is  the  drawing  of  practical  importance,  which  is 
based  on  the  perspective  diagram.  Fig.  20,  which  shows  the  mental 
conception  of  the  process  employed.  In  practical  work,  of  course. 
Fig.  21  alone  is  drawn,  and  it  is  constructed  by  geometrical  reason- 
ing deduced  from  the  mental  process  exhibited  by  Fig.  20. 

In  the  process  of  flattening  out  or  "  developing "  the  planes  of 
projection,  JJ  is  generally  considered  as  attached  or  hinged  to  the 
"  inclined  trace,"  XM  in  this  example.  In  Fig.  21  U  has  been 
revolved  about  XM,  bringing  it  into  the  plane  of  H,  the  trace  XN 
having  opened  out  to  two  lines.  N  separates  into  two  points  and  is 
marked  Nu  as  a  point  in  U  and  Nv  as  a  point  in  V,  analogous  to 
Yh  and  Ys  in  the  development  of  the  reference  planes.  The  space 
NuXNv,  like  YnOYs,  may  be  considered  as  construction  space. 


The  True  Length  of  a  Line  in  Space  29' 

26.  Fourth  law  of  Projection — that  for  Auxiliary  Plane,  U. — 

It  will  be  seen  from  Fig.  20  that  AAneAv  is  a  rectangle  and  that 
eAv  is  equal  io  AhA.  On  the  descriptive  drawing,  Fig.  21,  these 
two  lines,  eAv  and  eAn,  form  one  line  perpendicular  to  OX.  This 
is  in  accordance  with  the  first  law  of  projection  of  Art.  11. 

As  the  plane  \J  is  perpendicular  to  H  we  have  the  same  rela- 
tion there,  and  AAnkAu,  Fig.  20,  is  a  rectangle.  IcAu  is  therefore 
equal  to  AjiA,  and  in  the  development.  Fig.  21,  AJc  and  hA^  form 
one  line  AJcAn  perpendicular  to  XM. 

If  from  Au  and  Avy  Fig.  20,  perpendiculars  are  let  fall  upon  the 
intersection  of  U  and  V  (the  trace  XN)  they  will  meet  at  the 
common  point  I,  both  IcAJX  and  XlAvC  being  rectangles.  In  the 
descriptive  drawing.  Fig.  21,  AJ  is  perpendicular  to  XNu,  IJv  is 
the  arc  of  a  circle,  center  at  X,-  and  l^Av  is  perpendicular  to  XNv. 
The  following  law  of  projection  governs  the  position  of  Au  in 
the  plane  U- 

(4)  From  the  regular  projections  of  A  draw  perpendiculars 
to  the  traces  of  \].  These  lines  continued  into  the  field 
of  U  intersect  at  Au.  One  of  these  lines  is  carried  across 
the  construction  space  by  the  arc  of  a  circle  whose  center 
is  the  meeting  point  of  the  traces  of  U- 

27.  The  Graphical  Application  of  this  Law  to  a  Point. — The 
procedure  for  locating  the  projection  Au  on  the  descriptive  drawing, 
Fig.  21,  after  the  location  of  the  plane  U  has  been  determined,  is 
as  follows :  From  the  adjacent  projections  of  the  point  draw  lines 
perpendicular  to  the  traces  of  the  plane  U-  Continue  one  of  these 
lines  across  the  trace.  Swing  the  foot  of  the  other  perpendicular 
to  the  duplicated  trace,  and  continue  it  by  a  line  perpendicular  to 
this  trace  to  meet  the  line  first  mentioned.  Their  intersection  is 
the  projection  of  the  point  on  U-  In  Fig.  21,  this  requires  AiJcAu 
to  be  draw^n  perpendicular  to  XM,  and  the  line  AvUuAu  to  be 
traced  as  show*n. 

28.  The  True  Length  of  a  Line. — The  procedure  for  finding  the 
true  length  of  a  line  consists  in  first  drawing,  Fig.  21,  a  line  paral- 
lel to  one  of  the  projections  of  the  line  to  act  as  the  trace  of  the 
auxiliary  plane.  Where  this  trace  intersects  an  axis  of  projection 
perpendicular  lines  are  erected,  one  perpendicular  to  the  axis,  one 


30 


Engineering  Descriptive  Geometry 


perpendicular  to  the  trace.  These  lines  are  the  two  developed 
positions  of  the  other  trace  of  the  plane  \].  Then  locate  the  ex- 
tremities of  the  given  line  on  the  auxiliary  plane  HJ.  The  line 
joining  the  extremities  is  the  required  projection  of  the  line  on  \], 
and  is  equal  in  length  to  the  given  line. 

29.  Alternative  Method  of  Developing  the  Auxiliary  Plane,  HJ. — 
A  modification  of  this  construction  is  shown  in  the  descriptive 
drawing.  Fig.  22,  in  which  the  plane  JJ  has  been  revolved  about 
the  vertical  trace  XN  until  it  coincides  with  the  plane  V-  ^^ 
separates  into  two  lines,  XMh  and  Zil/«.  h^  of  Fig.  20,  becomes 
Jch  and  ku)  and  the  space  MhXMu  is  construction  space.     A  is  on 


Fig.  22. 


"a  horizontal  line  drawn  through  Av.  Ankh  is  perpendicular  to  XMn- 
Ichku  is  the  arc  of  a  circle  having  Z  as  a  center,  and  fc„A„  is  per- 
pendicular to  XMw    ^M  is  thus  located. 

This  method  of  development  of  the  planes  is  much  less  common 
in  practical  drawing  than  the  other,  because,  as  a  rule,  it  is  less 
convenient  than  the  first  method.  In  such  cases  as  occur  it  offers 
no  particular  difficulty.  Both  Figs.  21  and  22  are  solutions  of 
the  problem  of  finding  the  true  length  of  the  oblique  line  AB  by 
projection  on  an  auxiliary  inclined  plane,  U- 

30.  Alternative  Positions  of  the  Plane  U- — We  saw  that  the 
exact  position  of  the  plane  U?  so  long  as  it  remained  perpendicular 
to  H  and  parallel  to  AnBn,  was  left  to  choice  governed  by  practical 


The  True  Length  of  a  Line  in  Space  31 

considerations.  JJ  itself,  however,  may  be  taken  perpendicular  to 
V  and  parallel  to  AvBvj  or  it  may  be  taken  perpendicular  to  S  and 
parallel  to  AgBg.  To  get  an  entire  grasp  of  the  subject  the  student 
is  advised  to  trace  Fig.  21  on  thin  paper,  or  plot  it  on  coordinate 
paper,  points  A,  B,  and  X  being  (6,6,2),  (10,10,8)  and 
(11,  0,  0),  and  fold  the  figure  into  a  paper  box  diagram,  the  con- 
struction spaces  NuXNv  and  YhOYg  being  creased  in  the  middle 
and  folded  out  of  the  way.  Fig.  22  will  serve  equally  well.  The 
final  result  will  be  a  paper  box  exactly  similar  to  Fig.  20. 

The  variation  in  which  \J  is  perpendicular  to  V  may  be  plotted, 
passing  the  new  inclined  trace  of  U  (lettered  YM)  through  the 
point  (0,  0,  3)  parallel  to  AvBv.  Fold  this  figure  into  a  paper  box, 
the  paper  being  cut  along  a  line  YN  perpendicular  to  YM. 

The  other  variation  may  be  plotted  with  the  inclined  trace  of  U 
on  the  plane  S,  parallel  to  AgBa  and  passing  through  the  point 
(0,0,6)  (/,  of  Fig.  21).  Letter  this  trace  YsM  and  draw  YsN 
perpendicular  to  it,  inclining  up  to  the  right.  The  paper  must  be 
cut  on  this  line  to  enable  it  to  be  properly  folded. 

31.  The  Method  Applied  to  a  Plane  Figure.; — The  special  value 
of  this  use  of  the  auxiliary  plane  is  seen  when  one  operation  serves 
to  give  the  true  length  of  a  number  of  lines  at  once,  and  thus 
shows  a  whole  plane  figure  in  its  true  shape. 

In  Fig.  23  the  polygon  ABODE  is  shown  by  its  projection,  the 
point  A  alone  being  lettered.  It  is  noticeable  that  in  V  the  edges 
all  form  one  straight  line.  The  V  projector-planes  of  the  various 
edges  are  therefore  all  parts  of  the  same  plane,  and  the  polygon 
itself  is  a  plane  figure  placed  perpendicular  to  \.  It  may  be  said 
the  polygon  is  "  seen  on  edge  "  in  V- 

An  auxiliary  plane  JJ  has  been  taken  parallel  to  the  plane  of  the 
polygon,  and  therefore  perpendicular  to  V-  The  trace  XM  being 
parallel  to  the  V  projections  of  the  edges,  this  auxiliary  plane  serves 
to  show  the  true  length  of  all  the  edges  at  once.  The  projection  on 
HJ  is  the  true  shape  of  the  polygon  ABODE.  In  the  case  illustrated, 
the  U  projection  discloses  the  fact  that  the  polygon  is  a  regular 
pentagon,  a  fact  not  realized  from  the  regular  projections,  owing 
to  the  foreshortening  to  which  they  are  subject. 


32 


Engineering  Descriptive  Geometry 


This  figure  is  well  adapted  to  tracing  and  folding  into  a  paper 
box  diagram. 


Fig.  23. 


32.  The  True  Length  of  a  Line  by  Revolving  About  a  Projector. 

^-A  second  method  of  finding  the  true  length  of  a  line  seems  in  a 
way  simpler,  but  proves  to  be  of  much  less  value  in  practical  work. 
The  method  consists  in  supposing  an  oblique  line  AB  to  be  revolved 
about  a  projector  of  some  point  in  the  line  until  it  becomes  parallel 
to  one  of  the  planes  of  reference.  In  this  new  position  it  is  pro- 
jected to  the  reference  plane  as  of  its  true  length. 

In  Fig.  24  the  V  projector-plane  of  the  line  AB  has  been  shaded 
for  emphasis  (A  is  the  point  (1, 1,  5),  and  B  is  the  point  (5,  4,  2) ). 
The  projector  AAv  has  been  selected  at  will,  and  the  V  projector- 
plane  (of  which  the  line  AB  is  one  edge)  has  been  rotated  about 
AAv  as  an  axis  until  it  has  come  into  the  position  AvB'vB'A.  In 
its  new  position,  AB'  projects  to  ff  as  AnB'],.  This  is  the  true 
length  of  the  line.  During  its  rotation  the  point  B  has  moved  to 
B',  but  in  so  doing  it  has  not  revolved  about  A  as  its  center,  but 
about  the  point  h  on  AvA  extended.    hAv  is  equal  in  length  to  BBv. 


The  True  Length  of  a  Line  in  Space 


33 


Bi,  moves  to  B'v,  revolving  about  J.^  as  a  center.  In  Fig.  25,  the 
corresponding  descriptive  drawing,  the  original  projections  are 
shown  as  full  lines  and  the  projections  of  the  line  after  the  rota- 
tion has  occurred  are  shown  by  long  dashes. 

In  Yy  AvBv  swings  about  /!«  as  a  pivot  until  in  its  new  position 
AvB'v  it  is  parallel  to  OX.  In  H?  Bn  moves  in  a  line  parallel  to 
OX  (since  in  Fig.  24  the  motion  of  B  takes  place  entirely  in  the 
plane  of  IBB',  which  is  parallel  to  V)j  and  as  B\  must  be  verti- 
cally above  B'v  the  motion  terminates  where  a  line  drawn  vertically 


Fig.  24 


up  from  B'v  meets  the  horizontal  line  BhB\.  Joining  Ah  and  B'hy 
the  new  H  projection  is  the  true  length  of  the  given  line.  The  S 
projection  is  of  no  interest  in  this  case.  The  ff  and  V  projections 
of  Fig.  25  show  the  graphical  process  corresponding  to  the  theory 
of*  this  rotation.  In  V^  ^v  moves  to  B'v,  whence  a  .vertical  pro- 
jector meeting  a  horizontal  line  of  motion  from  Bh  determines  B'j,, 
the  new  position  of  Bh.  AnB'n  is  the  true  length  of  the  line.  The 
arrow-heads  on  the  broken  lines  make  these  steps  clear. 

33.  Variations  in  the  Method. — The  method  is  subject  to  wide 
variations.  The  same  projector-plane  AAvBvB,  Fig.  24,  revolving 
about  the  same  projector  AAv,  might  start  in  the  opposite  direction 
and  swing  to  a  position  parallel  to  S.  The  graphical  process  of 
Fig.  25  would  then  confine  itself  to  V  a^d  S  instead  of  V  and  H. 


34 


Engineeeixg  Descriptive  Geometry 


In  addition,  the  rotation  might  have  been  about  BBv  as  an  axis 
or  about  the  V  projector  of  any  point  in  AB  or  AB  extended. 
Finally,  the  H  projector-plane  or  the  S  projector-plane  might 
have  been  selected  and  made  to  revolve  into  position.  There  are 
six  distinct  varieties  of  the  process,  each  one  subject  to  great  modifi- 
cations. 

This  method  can  be  applied  to  a  plane  figure  which  appears  "  on 
edge''  in  one  of  the  regular  views.    ]n  Fig.  26  a  polygon  lies  in  a 


U:.^ N^/ 


V 


■"\\ 


4V+4 


3 


Fig.  26. 


plane  perpendicular  to  V-  There  are  two  varieties  of  the  process 
applicable  in  this  case.  Choosing  the  Y  projector  of  the  point  A 
for  the  axis  of  rotation,  the  whole  polygon  may  be  rotated  up  par- 
allel to  H,  thence  its  true  shape  projected  upon  f\,  or  it  may  be 
revolved  down  until  parallel,  to  §,  thence  its  true  shape 'projected 
upon  S-  Both  methods  are  shown,  though  of  course  in  practice  one 
at  a  time  should  be  enough. 

34.  A  Projector-Plane  Used  as  an  Auxiliary  Plane. — The  two 
processes  for  finding  the  true  length  of  a  line  differ  in  this  respect. 


The  True  Length  of  a  Line  in  Space  35 

In  one  the  line  is  projected  on  a  plane  which  is  revolved  into 
coincidence  with  one  of  the  reference  planes,  by  revolving  about  a 
line  in  that  reference  plane.  In  the  second  process,  a  projection 
plane  is  itself  revolved  about  a  projector,  that  is,  about  a  line 
perpendicular  to  one  reference  plane,  to  a  position  parallel  to  a 
second  reference  plane.  The  line  in  its  new  position  is  projected 
on  the  latter  plane. 

A  method  which  is  a  modification  of  the  first  process  is  in  many- 
cases  very  simple.  A  projector-plane  is  itself  used  as  an  auxiliary 
plane,  and  is  revolved  into  coincidence  with  the  plane  to  wliich  it  is 
perpendicular  by  rotation  about  its  trace  in  that  plane.  In  Fig.  23, 
for  example,  instead  of  passing  XM  parallel  to  AvCv,  AvCv  would  be 
extended  to  the  axis  of  X,  and  used  itself  for  the  inclined  trace  of 
the  auxiliary  plane.  XN  would  be  moved  to  the  right  and  other 
slight  modifications  made. 

As  in  the  second  method,  a  projector-plane  is  here  rotated;  but 
it  is  not  rotated  about  a  projector,  but  about  a  projection  (its 
trace),  and  the  real  similarity  of  the  process  is  with  the  first 
method,  that  of  the  auxiliary  plane  of  projection.  It  is  but  a 
special  case  of  this  kind. 

In  practical  drawing,  it  rarely  happens  that  one  of  the  projector- 
planes  can  be  thus  used  itself  with  advantage  as  an  auxiliary  plane 
of  projection.  It  leads  usually  to  an  overlapping  of  views  and  it 
will  not  be  found  so  useful  as  the  more  general  method. 

For  the  continuation  of  this  study,  all  these  methods  should  be 
at  the  students'  finger  ends. 

35.  The  True  Leng^th  of  a  Line  by  Constructing  a  Right  Tri- 
angle.— These  methods  of  finding  the  true  length  of  a  line  are 
generally  used  for  the  true  lengths  of  many  lines  in  one  operation, 
or  for  the  true  shape  of  a  plane  figure.  When  a  single  line  is 
wanted,  the  construction  of  a  right  triangle  from  lines  whose  true 
lengths  appear  on  the  drawing  is  sometimes  resorted  to.  In  Fig.  24 
the  triangle  ABh  is  a  right  triangle,  AB  being  the  hypotenuse  and 
AhB  the  right  angle.  In  the  descriptive  drawing,  Fig.  25,  AvBv 
is  equal  in  length  to  hB  of  Fig.  24,  and  AjJ)  is  easily  found,  equal 
to  Ah  of  Fig.  24.     These  lines  may  be  laid  off  at  any  convenient 


36  Engineering  Descriptive  Geometry 

place  as  the  sides  of  a  right  triangle,  and  the  hypotenuse  measured 
to  give  the  true  length  of  AB.  Mathematically  the  hypotenuse  is 
the  square  root  of  the  sum  of  the  squares  of  the  sides.  In  the  case 
illustrated  AiBv  is  5  (itself  the  square  root  of  A^c  +BvC  ,  ur 
^32  +  42  )  and  Anl)  is  3.  The  length  AB  is  therefore  VFTB^ 
=  V34  =  5.83. 

Problems  III. 

(For  use  with  wire-mesh  cage,  cross-section  paper,  or  blackboard.) 

30.  A  square  in  a  position  similar  to  the  pentagon  of  Fig. 
26  has  the  corners  A  (10,12,2),  B  (2,12,8),  C  (2,2.8)  and 
D  (10, 2,  2).    Find  its  true  shape  by  the  use  of  an  auxiliary  plane. 

31.  A  square  is  in  a  position  similar  to  the  pentagon  of  Fig.  23. 
The  comers  are  A  (9,3,3),  B  (9,13,3),  0  (3, 13, 11),  and 
D  (3,3,11).  Find  its  true  shape  by  revolving  into  a  plane  par- 
alel  to  fi. 

32.  Plot  the  triangle  A  (11,  3,  2),  B  (12,  6, 12),  C  (14, 12,  7). 
Find  its  true  shape  by  the  use  of  an  auxiliary  plane  perpendicular 
to  M. 

33.  Plot  the  triangle  A  (13,15,8),  B  (10,11,0),  C  (7,7,8). 
Find  the  true  shape  of  the  triangle  by  revolving  it  about  AA% 
until  in  a  plane  parallel  to  §.  Find  the  true  shape  by  projection 
on  a  plane  \],  perpendicular  to  ff,  whose  inclined  trace  passes 
through  the  point  (16,0,0).  (With  the  wire-mesh  cage  turn  the 
plane  S'  to  serve  for  this  auxiliary  plane  U-) 

34.  Same  with  triangle  A  (9,  7,  8),  B  (12, 11, 13),  C  (15, 15,  2). 

35.  Plot  the  right  triangle  A  (14,  4,  3),  B  (14, 10,  3),  C  (6,  4,  9). 
Revolve  it  about  BBv  into  a  plane  parallel  to  ff  and  project  its 
true  shape  on  J-J.  (With  the  wire-mesh  cage  put  markers  at  points 
A,  B,  C  and  C,  the  new  position  of  C.) 

36.  Plot  the  right  triangle  A  (9,  3,  6),  B  (9,  3,0),  C  (15, 11,  6). 
Eevolve  it  about  AB  until  in  a  plane  parallel  to  V  and  plot  C,  the 
new  position  of  the  vertex.  Revolve  it  about  the  same  axis  into  a 
plane  parallel  to  S?  and  plot  C",  the  new  position  of  the  vertex. 
(With  the  wire-mesh  cage  put  point  markers  at  A,  C,  C  and  C".) 


The  True  Length  of  a  Lixe  ix  Space  37 

37.  Plot  the  square  A  (14,8,2),  B  (ll,2,7i),  C  (11,14,74), 
D  (8,  8,  l'2f ).  The  diagonal  is  12  units  long.  Eevolve  the  square 
about  AAv  into  a  plane  parallel  to  fl,  and  project  its  true  shape  on 
J-J.  (With  the  wire-mesh  cage  put  point  markers  at  A,  B,  C,  D, 
B\  r,  and  D'.) 

38.  Plot  the  triangle  A  (12,2,14),  B  (2,2,14),  C  (7,7,2). 
Eevolve  it  about  AB  into  a  plane  parallel  to  V?  and  project  the  true 
shape  on  V-  (With  the  wire-mesh  cage  put  markers  at  points 
A,  B,  C  and  C  On  coordinate  paper  or  blackboard  show  the  true 
shape  by  projection  on  an  auxiliary  plane  \]  perpendicular  to  S? 
through  the  point  (0,8,0).) 

(For  use  on  coordinate  paper  or  blackboard,  not  wire-mesh  cage.) 

39.  The  triangle  A  (3,7,11),  B  (13,2,13),  C  (5,2,1)  is  a 
triangle  in  an  oblique  plane.  Find  its  true  shape  as  follows :  BC 
appears  at  its  true  length  in  V-  Draw  AvDv  perpendicular  to  BvCv. 
AD  is  an  oblique  line,  but  it  is  perpendicular  to  BC  since  its  V 
projector-plane  AAvDvD  is  perpendicular  to  BC.  Find  the  true 
length  of  AD  by  any  method.  On  V  extend  AvD^  to  Ev,  making 
DvEv  equal  to  the  true  length  of  AD.  EiByCv  is  the  true  shape  of 
the  triancrle  ABC. 


CHAPTER  IV. 

PLANE  STIRFACES  AND  THEIR  INTERSECTIONS  AND 
DEVELOPMENTS. 

36.  The  Omission  of  the  Subscripts,  h,  v,  and  s  — In  a  descriptive 
drawing  a  point  does  not  itself  appear  but  is  represented  by  its 
projections  on  the  reference  planes.  This  fact  has  been  emphasized 
in  the  previous  chapters.  In  the  more  complicated  drawings  which 
now  follow  it  will  save  time  and  will  prevent  overloading  the  figures 
with  lettering,  to  omit  the  subscripts  h,  v,  and  s,  and  to  refer  to  a 
point  and  its  projections  by  the  same  letter.  Thus  "  ^u  "  or  "  the 
point  A  in  V  ^^  8,re  expressions  which  call  attention  to  the  projec- 
tion of  A  on  V>  but  a  diagram  will  show  only  the  letter  A  at  that 
place.  If  at  any  time  it  is  necessary  to  be  more  precise  the  sub- 
scripts may  be  restored.  They  should  be  used  if  any  confusion  is 
caused  by  their  omission. 

If  the  projections  of  two  points  coincide,  it  is  sometimes  advis- 
able to  indicate  which  point  is  behind  the  other  in  that  view  by 
forming  the  letter  of  fine  dots.  Referring  back  to  Fig.  16,  the 
projections  of  A  arid  B  on  g  coincide.  On  this  system  subscripts 
are  omitted  and  the  letter  ^  (on  §  only)  is  formed  of  dots, 
as  in  Fig.  27. 

37.  Intersecting  Plane  Faces. — Many  pieces  of  machines  and 
structures  which  form  the  subjects  of  mechanical  drawings,  are 
pieces  all  of  whose  surfaces  are  portions  of  planes,  each  portion  or 
face  having  a  polygonal  outline. 

In  making  such  drawings  there  arise  problems  as'  to  the  exact 
points  and  lines  of  intersection,  which  can  be  solved  by  applying 
the  laws  of  projection  treated  of  in  the  preceding  chapters.  How 
these  intersections  are  determined  from  the  usual  data  will  now 
be  shown, 

38.  A  Pyramid  Cut  by  a  Plane. — As  a  simple  example  let  us 
suppose  that  it  is  required  to  find  where  a  plane  perpendicular  to 
V,  and  inclined  at  an  angle  of  30°  with  fi,  intersects  an  hex- 


Plane  Surfaces  and  Their  Intersections 


39 


agonal  pyramid  with  axis  perpendicular  to  Hi.  Fig.  27  is  the 
drawing  of  the  pyramid,  having  the  base  ABCDEF  and  vertex  P. 
The  cutting  plane  is  an  inclined  plane  such  as  we  have  used  for  an 
auxiliary  plane,  and  its  traces  are  therefore  similar  to  those  of  an 
auxiliary  plane.  KL  is  the  inclined  trace  on  V  and  KK'  and  LL' 
are  the  traces  parallel  to  the  axis  of  Y.  The  problem  is  to  find  the 
shape  of  the  polygonal  intersection  ahcdef  in  H  and  S,  and  its 
true  shape. 


The  method  of  solution  of  all  such  problems  is  to  take  into  con- 
sideration each  edge  of  the  pyramid  in  turn,  and  to  trace  the  points 
where  they  pierce  the  plane.  Thus,  the  edge  PA  pierces  the  given 
plane  at  a,  whose  projection  on  V  is  first  located;  for  the  given 
plane  is  seen  on  edge  in  V>  and  PA  cannot  pierce  the  plane  at  any 
other  point  consistent  with  that  condition,  a,  once  located  in  V> 
can  be  projected  horizontally  to  the  line  PA  in  S  and  vertically  to 
PA  in  H. 

The  true  shape  of  the  polygon  ahcdef  may  be  shown  on  an  aux- 
iliary plane,  U^  whose  traces  are  ZM  and  ZN.  In  Fig.  27  the 
projection  of  the  pyramid  on  HJ  is  incomplete.  As  it  is  only  to 
show  the  polygon  dbcdef  the  rest  of  the  figure  is  omitted. 


40 


Engineerixg  Descriptive  Geometry 


39.  Intersecting  Prisms. — As  an  example  of  somewhat  greater 
difficulty  let  it  be  required  to  find  the  intersection  of  two  prisms, 
one,  the  larger,  having  a  pentagonal  base,  parallel  to  fi  ;  and  the 
other  a  triangular  base,  parallel  to  §.  The  axes  intersect  at  right 
angles,  and  the  smaller  prism  pierces  the  larger. 


^- 


Hd 

r. 

H 
G' 

T 

—  V- 

■^  f' 

p  Jp^ 

fN 

k> 

The  known  elements  or  data  of  the  problem  are  shown  recorded 
as  a  descriptive  drawing  in  Fig.  28.  It  shows  the  projection  of 
the  pentagon  on  lil,  of  the  triangle  on  §,  and  of  the  axes  on  V- 
The  problem  is  to  complete  the  drawing  to  the  condition  of  Fig. 
29,  shown  on  a  larger  scale.  The  corners  of  the  pentagonal  prism 
are  ABODE f  and  A'B'C'D'E'f  and  its  axis  is  PP'.  The  corners 
of  the  triangular  prism  are  FGH  and  F'G'B'  and  its  axis  is  QQ'. 

40.  Points  of  Intersection. — The  general  course  in  solving  the 
problem  of  the  intersection  of  the  prisms  is  to  consider  each  edge 
of  each  prism  in  turn,  and  to  trace  out  where  each  edge  pierces  the 
various  plane  faces  of  the  other  prism.     When  all  such  points  of 


Plane  Suefaces  and  Their  Intersections  41 

intersection  have  been  determined,  they  are  joined  by  lines  to  give 
the  complete  line  of  intersection  of  the  prisms. 

To  determine  where  a  given  edge  of  one  prism  cuts  a  given  plane 
face  of  the  other  prism,  that  view  in  which  the  given  plane  face  is 
seen  as  a  line  only,  or  is  "seen  on  edge/'  as  is  said,  must  be  re- 
ferred to.  Taking  the  hexagonal  prism  first,  the  edges  AA',  CC, 
and  DD'  entirely  clear  the  triangular  prism,  as  is  disclosed  by  the 
plan  view  on  H  where  they  appear  "  on  end  "  or  as  single  points 
only.  They,  therefore,  have  no  points  of  intersection  with  the 
triangular  prism  and  in  V  and  §  these  lines  may  be  drawn  as 
nninterrupted  lines,  being  made  full  or  broken  according  to  the 
rule  at  the  end  of  Art.  5.  BB\  as  may  be  seen  in  H,  meets  the 
small  prism.  This  line  when  drawn  in  S,  where  the  plane  faces 
FF'G'G  and  FF'ITII  are  seen  on  edge,  meets  those  faces  at  h  and 
&'.  From  §  these  points  are  projected  to  V-  The  edge  BB'  con- 
sists really  of  two  parts,  Bh  and  h'B'.  EE'  meets  the  same  two 
faces  at  points  c  and  e'  determined  in  the  same  way. 

FF',  when  drawn  in  W,  is  seen  to  pierce  the  plane  face  AA'B'B 
at  /  and  AA'E'E  at  f.  These  points,  located  in  W,  are  projected 
vertically  down  to  V-  OG'  in  H  pierces  BB'C'C  at  g,  and  EE'D'D 
at  g'.  h  and  h'  on  the  line  HTF  are  similarly  determined  first  in 
H  and  are  projected  down  to  V- 

41.  Lines  of  Intersection. — Having  found  the  points  of  inter- 
section of  the  edges,  we  determine  the  lines  of  intersection  of  the 
plane  surfaces  by  considering  the  intersections  of  plane  with  plane, 
instead  of  line  with  plane.  BB'  is  one  line  of  the  plane  AA'B'B, 
and  pierces  the  plane  FF'G'G  (seen  on  edge  in  S)  at  &.  h  is  there- 
fore a  point  of  both  planes.  FF'  is  a  line  of  the  plane  FF'G'G,  and 
it  pierces  the  plane  AA'B'B  (seen  on  edge  in  IHI)  at  /,  /  is  also  a 
point  common  to  both  planes.  Since  these  two  points  are  in  both 
planes,  they  are  points  on  the  line  of  intersection  of  the  two  planes. 
We  therefore  connect  h  and  /  by  a  straight  line  in  \,  but  do  not 
extend  it  beyond  either  point  because  the  planes  are  themselves 
limited. 

By  the  same  kind  of  reasoning  h  and  </  are  found  to  be  points 
common  to  BB'C'C  and  FF'G'G,  and  are  therefore  joined  by  a 
straight  line,  hg  in  V-    9^^  also  is  the  line  of  intersection  of  two 


42 


Engineeeing  Descriptive  Geometry 


planes,  and  the  student  should  follow  for  himself  the  full  process 
of  reasoning  which  proves  it.  e,  f,  and  g'  are  points  similar  to  l,  f, 
and  g.  Since  the  original  statement  required  the  triangular  prism 
to  pierce  the  pentagonal  one,  gg',  ff,  and  liTi'  are  joined  by  broken 
lines  representing  the  concealed  portions  of  the  edges  GG' ,  FF',  and 
Eir  of  the  small  prism.  Had  it  been  stated  that  the  object  was 
one  solid  piece  instead  of  two  pieces,  these  lines  would  not  exist  on 
the  descriptive  drawing. 

42.  Use  of  an  Auxiliary  Plane  of  Projection. — To  find  the  inter- 
section of  solids  composed  of  plane  faces,  it  is  essential  to  have 


Fig.  30. 


views  in  which  the  various  plane  faces  are  seen  on  edge.    To  obtain 
such  views,  an  auxiliary  plane  of  projection  is  often  needed. 

Fig.  30  shows  the  data  of  a  problem  which  requires  the  auxiliary 
view  on  \]  in  order  to  show  the  side  planes  of  the  triangular  prism 
"on  edge.''  (These  planes  are  oblique,  not  inclined,  and  therefore 
do  not  appear  "on  edge"  on  any  reference  plane.)  Fig.  31  shows 
the  complete  solution,  the  object  drawn  being  one  solid  piece  and 
not  one  prism  piercing  another  prism,  h  and  d  are  located  by  the 
use  of  the  view  on  U-  Tn  this  case  and  in  many  similar  cases  in 
practical  drawing,  the  complete  view  on  IjJ  need  not  be  constructed. 


Plane  Surfaces  and  Their  Intersections 


43' 


The  use  of  U  is  only  to  give  the  position  of  b  and  d,  which  are  then 
projected  to  V-  The  construction  on  HJ  of  the  square  ends  of  the 
square  prism  are  quite  superfluous,  and  would  be  omitted  in  prac- 
tice. In  fact,  the  view  U  would  be  only  partially  constructed  in 
pencil,  and  would  not  appear  on  the  finished  drawing  in  ink. 

After  the  method  is  well  understood,  there  will  be  no  uncertainty 
as  to  how  much  to  omit. 

43.  A  Cross-Section. — In  practical  drawing  it  often  occurs  that 
useful  information  about  a  piece  can  be  given  by  imagining  it  cut 
by  a  plane  surface,  and  the  shape  of  this  plane  intersection  drawn. 
In  machine  drawing,  such  a  section  showing  only  the  material 
actually  cut  by  the  plane  and  nothing  beyond,  is  called  a  "  cross^ 


H 

«» 

X 

'— : — 1 

1 — ' 

m 

■        1 

n.       ' 1 

n 

1  1 

X 

0 
Z 

32. 

1                               "V 

i  11  i 

A 

1" 

PV 

1    1^ 

^     %^ 

^^ 

V 

Fig. 

s 

section."  In  other  branches  of  drawing  other  names  for  the  same 
kind  of  a  section  are  used.  The  "  contour  lines  "  of  a  map  are  of 
this  nature,  as  well  as  the  "  water  lines "  of  a  hull  drawing  in 
Naval  Architecture. 

44.  Sectional  Views. — ^The  cross-section  is  used  freely  in  ma- 
chinery drawing,  but  a  "  sectional  view,"  which  is  a  view  of  a  cross- 
section,  with  all  those  parts  of  the  piece  which  lie  beyond  the  plane 
of  the  section  as  well,  is  much  more  common. 

These  sectional  views  are  sometimes  made  additional  to  the 
regular  views,  but  often  replace  them  to  some  extent.    Fig.  32  is  a 


44  Engineering  Descriptive  Geometry 

good  example.  It  represents  a  cast-iron  structural  piece  shown  by 
plan,  and  two  sectional  views.  The  laws  of  projection  are  not 
altered,  but  the  views  bear  no  relation  to  each  other  in  one  respect. 
One  view  is  of  the  whole  piece,  one  is  of  half  the  piece,  and  one  is 
of  three-quarters  of  it.  The  amount  of  the  object  imagined  to  be 
cut  away  and  discarded  in  each  view  is  a  matter  of  independent 
choice. 

In  the  example  the  projection  on  V  is  a  view  of  half  of  the 
piece,  imagining  it  to  have  been  cut  en  a  plane  shown  in  lil  by  the 
line  mn.  The  half  between  mn  and  OX  has  been  discarded,  and  the 
drawing  shows  the  far  half.  The  actual  section,  the  cross-section 
on  the  line  mn,  is  an  imaginary  surface,  not  a  true  surface  of  the 
object,  and  it  is  made  distinctive  by  "hatching."  This  hatching 
is  a  conventional  grouping  of  lines  which  show  also  the  material 
of  which  the  piece  is  formed.  For  this  subject,  consult  tables  of 
standards  as  given  in  works  on  Mechanical  Drawing.  This  pro- 
jection on  V  is  not  called  a  "  Front  Elevation,"  but  a  "  Front  Ele- 
vation in  Section,"  or  a  "  Section  on  the  Front  Elevation." 

The  view  projected  on  S  is  called  a  "  Side  Elevation,  Half  in 
Section,"  or  a  "  Half-Section  on  the  Side  Elevation."  Since  a 
eection  generally  means  a  sectional  view  of  the  object  with  half 
removed,  a  half-section  means  a  view  of  the  object  wath  one-quarter 
removed.  If,  in  H,  the  object  is  cut  by  a  plane  whose  trace  is  np 
and  another  whose  trace  is  pq,  and  the  N".  E.  comer  of  the  object 
is  removed,  it  will  correspond  to  the  condition  of  the  object  as  seen 

inS- 

Sections  are  usually  made  on  the  center  lines,  or  rather  on  central 
planes  of  the  object.  When  strengthening  ribs  or  "  webs  "  are  seen 
in  machine  parts,  it  is  usual  to  take  the  plane  of  the  section  just 
in  front  of  the  rib  rather  than  to  cut  a  rib  or  web  which  lies  on  the 
central  plane  itself.  This  position  of  the  imaginary  saw-cut  is 
selected  rather  than  the  adjacent  center  line. 

AYhen  the  plane  of  a  section  is  not  on  a  center  line,  or  adjacent 
to  one,  its  exact  location  should  be  marked  in  one  of  the  views  in 
which  it  appears  "  on  edge,"  and  reference  letters  put  at  the  ex- 
tremities.   The  section  is  then  called  the  "  section  on  the  line  mn/* 

The  passing  of  these  section  planes  causes  problems  in  intersec- 
tion to  arise,  which  are  similar  to  those  treated  in  Articles  37-42. 


Plane  Surfaces  and  Their  Intersections 


45 


45.  Development  of  a  Prism. — It  is  often  desired  to  show  the 
true  shape  of  all  the  plane  faces  of  a  solid  object  in  one  view, 
keeping  the  adjacent  faces  in  contact  as  much  as  possible.  This 
is  called  developing  the  surface  on  a  plane,  and  is  particularly 
useful  for  all  objects  made  of  sheet-metal,  as  the  development  forms 
a  pattern  for  cutting  the  metal,  which  then  requires  only  to  be  bent 
into  shape  and  the  edges  to  be  joined  or  soldered. 


Development  is  a  process  already  applied  to  the  planes  of  pro- 
jection themselves  when  these  planes  were  revolved  about  axes  until 
all  coincided  in  one  plane.  The  same  operation  applied  to  the 
surfaces  of  the  solid  itself  produces  the  development. 

The  two  prisms  of  Fig.  29  afford  good  subjects  for  development. 
Fig.  33  shows  the  developed  surface  of  the  triangular  prism,  the 
lines  g-g  and  g'-g'  showing  the  lines  of  intersection  with  the  other 
prism.  In  this  figure  it  is  considered  that  the  surface  of  the 
triangular  prism  is  cut  along  the  lines  GG\  OF,  G'F',  GH,  and 


46  Engineering  Descriptive  Geometry 

O'H' ',  and  the  four  outer  planes  unfolded,  using  the  lines  bound- 
ing FF'H'H  as  axes,  until  the  entire  surface  is  flattened  out  on  the 
plane  of  FF'H'H. 

Fig.  34  shows  the  development  of  the  large  prism  of  Fig.  29, 
with  the  holes  where  the  triangular  prism  pierces  it  when  the  two 
are  assembled.  The  surface  of  the  prism  is  cut  on  the  line  AA', 
and  on  other  lines  as  needed,  and  the  surfaces  are  flattened  out  by 
unfolding  on  the  edges  not  cut. 

The  construction  of  these  developments  is  simple,  since  the  sur- 
faces are  all  triangles  or  pentagons  whose  true  shapes  are  given; 
or  are  rectangles,  the  true  length  of  whose  edges  are  already  known. 
.  In  Fig.  33  the  distances  Og,  G'/,  Ff,  F'f  are  taken  directly 
from  V  in  Fig.  29.  The  points  h  and  e  are  plotted  as  follows: 
The  perpendicular  distance  hi  to  the  line  GF  is  taken  from  V> 
Fig.  29,  and  Gl  is  taken  from  Gl  in  S,  Fig.  29.  The  other  points 
are  plotted  in  the  same  manner. 

46.  Development  of  a  Pyramid. — Fig.  35  shows  the  development 
of  the  point  of  the  pyramid,  Fig.  27,  cut  off  by  the  intersecting 


plane  whose  trace  is  KL.  The  base  is  taken  from  the  projection 
on  IUj  where  its  true  shape  is  given.  Each  slant  side  must  have  its 
true  shape  determined,  either  as  a  whole  plane  figure  (Art.  31), 
or  by  having  all  three  edges  separately  determined  (Art.  28  or 
Art.  32).  In  this  case  Pa  and  Pd  are  shown  in  true  length  in  V? 
Fig.  27,  and  it  is  only  necessary  to  determine  the  true  lengths  of 
Pb  and  Pc  (or  their  equivalents  Pf  and  Pe)  to  have  at  hand  all  the 
data  fox  laying  out  the  development.  The  face  Pef  may  be  con- 
veniently shown  in  its  true  shape  on  an  auxiliary  plane  "W?  Fig.  27, 
perpendicular  to  §  and  cutting  §  in  a  trace  YgN  as  shown. 


Plane  Surfaces  and  Their  Intersections  47 

Problems  IV. 

(For  use  with  wire-mesh  cage,  or  on  cross-section  paper  or 
blackboard. ) 

40.  Plot  the  projections  of  the  points  A  (9,  3, 16),  B  (6,  3, 16), 
C  (6,  8, 16),  D  (9,  8, 16),  and  E  (0,  3,  4),  i^  (0,  3,  8),  G  (0,  8,  8), 
H  (0,8,4).  Join  the  projections,  A  to  E,  B  to  F,  C  to  G,  etc. 
(With  the  wire-mesh  cage  use  stiff  wire  to  represent  the  lines  AE, 
BF,  etc.)  Show  how  to  find  the  true  shape  of  every  plane  surface 
of  the  solid  (a  prism)  thus  formed.  On  cross-section  paper  or  on 
blackboard  show  how  to  draw  the  development  of  the  surface  of 
the  solid. 

41.  Same  as  Problem  40,  with  points  A  (10,  8,  0),  B  (8, 10,  0), 
C  (12, 14,  0),  D  (14, 12,  0)  on  fi  and  E  (10,  8, 16),  F  (6, 12, 16), 
O  (8, 14, 16),  H  (12, 10, 16)  on  H'.  (In  developing  the  surface, 
find  the  true  shape  of  the  quadrilateral  BFGC  by  dividing  it  into 
two  triangles  by  a  diagonal  BG  whose  true  length  will  appear  on  g. 
Divide  CGHD  by  the  diagonal  CH.) 

42.  Draw  the  tetrahedron  whose  four  comers  are  A  (16,2,13), 
B  (6,  2, 13),  C  (11, 14, 13)  and  D  (11,  7. 1).  It  is  intersected  by 
a  plane  perpendicular  to  V  cutting  V  in  a  trace  passing  through 
the  origin,  making  an  angle  of  30°  with  OX.  Draw  the  trace  of 
the  plane  on  V*  Where  are  its  traces  on  fj  and  S  ?  Show  the  H 
and  S  projections  of  the  intersection  of  the  plane  and  tetrahedron. 

43.  A  solid  is  in  the  form  of  a  pyramid  whose  base  is  a  square  of 
10  units,  and  whose  height  is  8  units.  The  corners  are  A  ( 16,  2, 10) , 
B  (10,2,2),  C  (2,2,8)  and  D  (8,2,16)  and  the  vertex 
E  (9,10,9).  It  is  intersected  by  a  plane  perpendicular  to  H^ 
whose  trace  on  Hil  passes  through  the  origin  making  an  angle  of 
30°  with  OX,  Draw  the  V  and  S  projections  of  the  intersection 
of  the  pyramid  and  plane.  Where  is  the  trace  of  the  cutting  plane 
on  V? 

44.  A  plane  H'  is  parallel  to  H  at  a  distance  of  16  units,  A 
square  prism  has  its  base  in  H-  The  corners  are  ^  (8,2,0), 
B  (3,  7,  0),  C  (8, 12,  0),  D  (13,  7,  0).  Its  other  base  is  in  H',  the 
comers  A',  B',  etc.,  having  the  same  x  and  y  coordinates  as  above, 
and  the  z  coordinates  16. 


48.  Engineering  Descriptive  Geometry 

A  plane  S'  is  parallel  to  S  at  a  distance  of  IG.  A  triangular  prism 
has  its  base  in  S,  points  E  (0,  5,  8),  F  (0,  13,  2),  G  (0,  13,  1-1)  ; 
and  its  other  base  in  S',  points  E',  F',  G'  having  x  coordinates  16, 
and  y  and  z  coordinates  unchanged.  Make  the  drawing  of  the  in- 
tersecting prisms  considering  the  triangular  prism  to  be  solid  and 
parts  of  the  square  prism  cut  away  to  permit  the  triangular  one  to 
pass  through. 

(For  use  on  cross-section  paper  or  blackboard,  not  wire-mesh 
cage.) 

45.  A  sheet-iron  coal  chute  connects  a  square  port,  A  (2,4,2), 
B  (2,12,2),  C  (2,12,10),  D  (2,4,10),  with  a  square  hatch, 
E  (14,  6, 16),  F  (14, 10, 16),  G  (10, 10, 16),  E  (10,  6, 16).  The 
corners  form  lines  AE,  BF,  CG,  DH  and  the  side  plates  are  bent 
on  the  lines  AH  and  BG.    Draw  the  development  of  the  surface. 

46.  Draw  the  development  of  the  tetrahedron  of  Problem  42 
with  the  line  of  intersection  marked  on  it. 

47.  Draw  the  development  of  the  pyraniid  of  Problem  43  with 
the  line  of  intersection  marked  on  it. 

48.  Draw  the  development  of  the  square  prism  of  Problem  44 
with  the  line  of  intersection  marked  on  it. 

49.  Draw  the  development  of  the  triangular  prism  of  Problem 
44  with  the  line  of  intersection  marked  on  it. 


CHAPTEE  Y. 

CURVED  LINES. 

47.  The  Simplest  Plane  Curve,  the  Circle. — The  geometrical 
natures  of  the  common  curves  are  supposed  to  be  understood.  De- 
scriptive Geometry  treats  of  the  nature  of  their  orthographic  pro- 
jections. The  curves  now  considered  are  plane  curves,  that  is, 
every  point  of  the  curve  lies  in  the  same  plane.  It  is  natural, 
therefore,  that  the  relation  of  the  plane  of  the  curve  to  the  plane  of 
projection  governs  the  nature  of  the  projection. 


Plane 
of  the 
^  Circle 

^^ 

f 

^=^^ 

Fig.  36. 


Fig.  37. 


The  simplest  plane  curve  is  a  circle.  Figs.  36  and  37  show  the 
three  forms  in  which  it  projects  upon  a  plane.  In  Fig.  36,  a  per- 
spective drawing,  we  have  a  circle  projected  upon  a  parallel  plane 
of  projection  (that  in  the  position  customary  for  V).  The  pro- 
jectors are  of  equal  length  and  the  projection  is  itself  a  circle  ex- 
actly equal  to  the  given  circle. 

On  a  second  plane  of  projection  (that  in  the  position  of  §)  per- 
pendicular to  the  plane  of  the  circle  the  projection  is  a  straight 
line  equal  in  length  to  the  diameter  of  the  circle,  AC.  The  pro- 
jectors for  this  second  plane  of  projection  form  a  projector-plane. 


60 


Engineering  Descriptive  Geometry 


In  Fig.  37  the  circle  is  in  a  plane  inclined  at  an  angle  to  the 
plane  of  projection.  The  projectors  are  of  varying  lengths.  There 
must  be  one  diameter  of  the  circle,  however,  that  marked  AC, 
which  is  parallel  to  the  plane  of  projection.  The  projectors  from 
these  points  are  of  equal  length,  and  the  diameter  AC  appears  of 
its  true  length  on  the  projection  as  AvCv. 

The  diameter  BD  at  right  angles  to  AC,  has  at  its  extremity  B 
the  shortest  projector,  and  at  the  extremity  D  the  longest  projector. 
On  the  projection,  BD  appears  greatly  foreshortened  as  BvDcy 
though  still  at  right  angles  to  the  projection  oi  AC  and  bisected 
by  it. 


1 

X 

/ 

^^ 

w- 

\ 

^X 

t 

— C7 

_  .  •* 

f 

-^ 

l\ 

B^ 

i 

■-, 

X 

1 

\0 

i 

! 

L 

3 

i      !  ^ 

i             ! 

A 

BD 

c 

CA           J> 

z 

Fig 

38. 

The  true  shape  of  the  projection  is  an  ellipse,  of  which  A^Cv  is 
the  major  axis  and  BJ)v  is  the  minor  axis.  No  matter  at  what 
angle  the  plane  of  projection  lies,  the  projection  of  a  circle  is  an 
ellipse  whose  major  axis  is  equal  to  the  diameter  of  the  circle. 

For  convenience  the  two  planes  of  projection  in  Fig.  36  have 
been  considered  as  V  and  S,  and  the  projections  lettered  accord- 
ingly. The  plane  of  projection  in  Fig.  37  has  been  treated  as  if 
it  were  V?  and  the  ellipse  so  lettered.  It  must  be  remembered  that 
the  three  forms  in  which  the  circle  projects  upon  a  plane,  as  a 
circle,  as  a  line,  and  as  an  ellipse,  cover  all  possible  cases,  and  the 
relations  between  the  plane  of  the  circle  and  the  plane  of  projec- 
tion shown  in  the  two  figures  are  intended  to  be  perfectly  general 
and  not  confined  to  V  and  §  alone. 


Curved  Lines 


51 


48.  The  Circle  in  a  Horizontal  or  Vertical  Plane. — Passing  now 
to  the  descriptive  drawing  of  a  circle,  the  simplest  case  is  that  of 
a  circle  which  lies  in  a  plane  parallel  to  fl,  V  or  §.  The  projec- 
tions are  then  of  the  kind  shown  in  Fig.  36,  two  projections  being 
lines  and  one  the  true  shape  of  the  circle.  Fig.  38  shows  the  case 
for  a  circle  lying  in  a  horizontal  plane.  The  true  shape  appears  in 
Hi.  The  V  projection  shows  the  diameter  AC,  the  S  projection 
shows  the  diameter  BD. 


Fig.  39. 


49.  The  Circle  in  an  Inclined  Plane. — Fig.  39  shows  the  circle 
lying  in  an  inclined  plane,  perpendicular  to  Vj  and  making  an 
angle  of  60°  with  ffil.  The  V  projectors,  lying  in  the  plane  of  the 
circle  itself,  form  a  projector-plane  and  the  V  projection  is  a 
straight  line  equal  to  a  diameter  of  the  circle.  As  the  plane  of  the 
circle  is  oblique  to  H  and  §,  these  projections  on  ff  and  S  are 
ellipses  Avhose  major  axes  are  equal  to  the  diameter  of  the  circle. 
Of  course,  for  any  point  of  the  curve,  as  P,  the  laws  of  projection 
hold,  as  is  indicated.    The  true  shape  of  the  curve  can  be  shown  by 


52 


Engineerixg  Descriptive  Geometry 


projection  on  any  plane  parallel  to  the  plane  of  the  circle.  It  is 
here  shown  on  the  auxiliary  plane  HJ,  taken  as  required.  If  the 
drawing  were  presented  with  projections  H,  V  and  g,  as  shown, 
one  might  at  first  suspect  that  it  represented  an  ellipse  and  not  a 
circle ;  but,  if  a  number  of  points  were  plotted  on  \],  the  existence 
of  a  center  0'  could  be  proved  by  actual  test  with  the  dividers. 

50.  The  Circle  in  an  Oblique  Plane. — ^AVhen  a  circle  is  in  an 
oblique  plane,  all  three  projections  are  ellipses,  as  in  Fig.  40.  The 
noticeable  feature  is  that  the  three  major  axes  are  all  equal  in 
length. 


When  an  ellipse  is  in  an  oblique  plane,  its  three  projections  are 
also  ellipses,  but  the  major  axes  will  be  of  unequal  lengths.  The 
proof  of  this  fact  must  be  left  until  later.  The  fact  that  the  three 
projections  have  their  major  axes  equal  must  be  taken  at  present  as 
sufficient  evidence  that  the  curve  itself  is  a  circle. 

51.  The  Ellipse:  Approximate  Representation. — The  ellipse  is 
little  used  as  a  shape  for  machine  parts.  It  appears  in  drawings 
chiefly  as  the  projection  of  a  circle.  Some  properties  of  ellipses 
are  very  useful  and  should  be  studied  for  the  sake  of  reducing  the 
labor  of  executing  drawings  in  which  ellipses  appear.. 

An  approximation  to  a  true  ellipse  by  circular  arcs,  known  as  the 
"  draftsman's  ellipse/'  may  be  constructed  when  the  major  axis  2a 
and  the  minor  axis  2h,  Fig.  41,  of  an  ellipse  are  known. 


Curved  Lines 


53 


The  steps  in  the  process  are  shown  in  Fig.  41.,  The  center  of  the 
ellipse  is  at  0.  The  major  axis  is  AC,  equal  to  2a.  The  minor 
axis  is  DB,  equal  to  2&.  From  C,  one  end  of  the  major  axis,  lay 
off  CE,  equal  to  I.  The  point  £'  is  at  a  distance  equal  to  a—h  from 
0  and  at  a  distance  equal  to  2a — h  from  A.  This  last  distance  is 
the  radius  of  a  circular  arc  which  is  used  to  approximate  to  the 
flat  sides  of  the  ellipse.  It  may  be  called  the  "  side  arc."  Setting 
the  compass  to  the  distance  AE  and  using  D  and  B  as  centers, 
points  H  and  0  are  marked  on  the  minor  axis,  extended,  for  use 
as  centers  for  the  "  side  arcs.'*  These  arcs  are  now  drawn  (passing 
through  the  points  D  and  5),  as  shown  in  the  2nd  stage  of  the 
process. 


\ 

\  5^  5fa^e 

THE 


3rd  Stage 


DRAFTSMAN'S   ELLIPSE." 
Fig,  41. 


By  use  of  the  bow  spacer,  the  distance  OE  is  bisected  and  the 
half  added  to  itself,  giving  the  point  F  (distant  f  (^— &)  from  0). 
F  is  the  center  of  a  circular  arc  which  approximates  to  the  end  of 
the  true  ellipse.  With  F  as  center,  and  FC  as  radius,  describe  this 
arc.  If  this  work  is  accurate,  this  "  end  arc "  will  prove  to  be 
tangent  to  the  side  arcs  already  drawn,  as  shov/n  in  the  3rd  stage 
of  the  process.  If  desired,  the  exact  point  of  tangency  of  the  two 
arcs,  K,  may  be  found  by  joining  the  centers  H  and  F  and  extend- 
ing the  line  to  K.  F  is  s^vung  about  0  as  center  by  compass  or 
dividers  to  F',  for  the  center  of  the  other  ^^end  arc."  In  inking 
such  an  ellipse,  the  arcs  must  be  terminated  exactly  at  the  points 
of  tangency,  K  and  the  three  similar  points. 

This  method  is  remarkably  accurate  for  ellipses  whose  minor 
5 


54 


Enginei;eing  Descriptive  Geometry 


axes  are  at  least  two-thirds  the  length  of  their  major  axes.  It 
should  always  be  used  for  such  wide  ellipses,  and  if  the  character 
of  the  drawing  does  not  require  great  accuracy,  it  may  be  used 
even  when  the  minor  axis  is  but  half  the  length  of  the  major  axis. 
For  all  narrow  ellipses,  exact  methods  of  plotting  should  be  used. 
52.  The  Ellipse:  Exact  Representation. — The  true  and  accurate 
methods  of  plotting  an  ellipse  are  shown  in  Figs.  42,  43,  and  44. 
Fig.  42  is  a  convenient  method  when  the  major  axis  AC  and  minor 
axis  BD  are  given,  bisecting  each  other  at  0.  Describe  circles  with 
centers  at  0,  and  with  diameters  equal  to  AC  and  BD.  From  0 
draw  ani/  radial  line.  From  the  point  where  this  radial  line  meets 
the  larger  circle  draw  a  vertical  line,  and  from  the  point  where  it 
cuts  the  smaller  circle  draw  a  horizontal  line.     Where  these  lines 


Fig.  42. 


Fig.  43 


meet  at  P  is  located  a  point  on  the  ellipse.  By  passing  a  large 
number  of  such  radial  lines  sufficient  points  may  be  found  between 
D  and  C  to  fully  determine  the  quadrant  of  the  ellipse.  Having 
determined  one  quadrant,  it  is  generally  possible  to  transfer  the 
curve  by  the  pearwood  curves  with  less  labor  than  to  plot  each 
quadrant. 

With  the  same  data  a  second  metho(L  Fig.  43,  is  more  convenient 
for  work  on  a  large  scale  when  the  T-square,  beam  compass,  etc., 
are  not  available. 

Construct  a  rectangle  using  the  given  major  and  minor  axes  as 
center  lines.  Divide  DE  into  any  number  of  equal  parts  (as  here 
sliown,  4  parts),  and  join  these  points  of  division  with  C.  Divide 
DO  into  the  same  number  of  equal  parts  (here,  4).  From  A 
draw  lines  through  these  last  points  of  division,  extending  them  to 
the  first  system  of  lines  intersecting  the  first  of  the  one  system  with 


Curved  Lines  55 

the  first  of  the  other,  the  second  with  the  second,  etc.  These  inter- 
sections, 1,  2,  3,  are  points  on  the  ellipse. 

The  third  method,  an  extension  or  generalization  of  the  second,  is 
very  useful  when  an  ellipse  is  to  be  inscribed  in  a  parallelogram,  the 
major  and  minor  axes  being  unknown  in  direction  and  magnitude. 
Lettering  the  parallelogram  A'B'C'D'  in  a  manner  similar  to  the 
lettering  in  Fig.  43,  the  method  is  exactly  the  same  as  before,  D'E' 
and  D'O  being  divided  into  an  equal  number  of  parts  and  the  lines 
drawn  from  C  and  A\  The  actual  major  and  minor  axes,  indicated 
in  the  figure,  are  not  determined  in  any  manner  by  this  process. 

53.  The  Helix. — The  curve  in  space  (not  a  plane  curve)  which 
is  most  commonly  used  in  machinery,  is  the  helix.  This  curve  is 
described  by  a  point  revolving  uniformly  about  an  axis  and  at  the 
same  time  moving  uniformly  in  the  direction  of  that  axis.  It  is 
popularly  called  a  "  cork-screw  ^'  curve,  or  "  screw  thread,"  or  even, 
quite  incorrectly,  a  "  spiral." 

The  helix  lies  entirely  on  the  surface  of  a  cylinder,  the  radius  of 
the  cylinder  being  the  distance  of  the  point  from  the  axis  of  rota- 
tion, and  the  axis  of  th?  cylinder  the  given  direction. 

Fig.  45  represents  a  cylinder  on  the  surface  of  which  a  moving 
point  has  described  a  helix.  Starting  at  the  top  of  the  cylinder,  at 
the  point  marked  0,  the  point  has  moved  uniformly  completely 
around  the  cylinder  at  the  same  time  that  it  has  moved  the  length 
of  the  cylinder  at  a  uniform  rate.  The  -circumference  of  the  top 
circle  of  the  cylinder  has  been  divided  into  twelve  equal  parts  by 
radii  at  angles  of  30°,  the  apparent  inequality  of  the  angles  being  due 
to  the  perspective  of  the  drawing.  The  points  of  division  are  marked 
from  0  to  11,  point  12  not  being  numbered,  as  it  coincides  with 
point  0.  The  length  of  the  cylinder  is  divided  into  twelve  equal 
parts  on  the  vertical  line  showing  the  numbers  from  0  to  12,  and 
at  each  point  of  division  a  circle,  parallel  to  the  top  base,  is  de- 
scribed about  the  cylinder.  The  helix  is  the  curve  shown  by  a 
heavy  line.  From  point  0,  which  is  the  zero  point  of  both  move- 
ments, the  first  twelfth  part  of  the  motion  carries  the  point  from  0 
to  1  around  the  circumference,  and  from  0  to  1  axially  downward, 
at  the  same  time.  The  true  movement  is  diagonally  across  the 
curved  rectangle  to  the  point  marked  1  on  the  helix.     This  move- 


53 


ExGiNEERiNG  Descriptive  Geometry 


ment  is  continued  step  by  step  to  the  points  2,  3,  etc.  In  the  posi- 
tion chosen  in  Fig.  45,  points  0,  1,  2,  3,  4,  12  are  in  full  view, 
points  5  and  11  are  on  the  extreme  edges,  and  the  intermediate 


Fig.  45. 


points  (from  6  to  10)  are  on  the  far  side  of  the  cylinder.  The 
construction  lines  for  these  latter  points  have  been  omitted,  in  order 
to  keep  the  figure  clear. 


Curved  Lines  57 

54.  Projections  of  the  Helix. — The  projection  of  this  curve  on  a 
plane  parallel  to  the  axis  of  the  C3'linder  is  shown  to  the  left.  The 
circles  described  about  the  cylinder  become  equidistant  parallel 
straight  lines.  The  axial  lines  remain  straight  but  are  no  longer 
equally  spaced,  and  the  curve  is  a  kind  of  continuous  diagonal  to 
the  small  rectangles  formed  by  these  lines  on  the  plane  of  projec- 
tion. 

The  projection  of  the  helix  on  any  plane  perpendicular  to  the 
axis  of  the  cylinder  is  a  circle  coinciding  with  the  projection  of  the 
cylinder  itself.  The  top  base  is  such  a  plane  and  on  it  the  projec- 
tion of  the  helix  coincides  with  the  circumference  of  the  base. 

55.  Descriptive  Drawing  of  the  Helix. — The  typical  descriptive 
drawing  of  a  helix  is  shown  in  Fig.  46.  The^axis  of  the  cylinder  is 
perpendicular  to  ff,  and  the  top  base  is  parallel  to  ff.  The  helix 
in  IHI  appears  as  a  circle.  In  V-it  appears  as  on  the  plane  of  pro- 
jection in  Fig.  45,  but  this  view  is  no  longer  seen  obliquely  as  is 
there  represented. 

This  V  projection  of  the  helix  is  a  plane  curve  of  such  import- 
ance as  to  receive  a  separate  name.  It  is  called  the  "sinusoid." 
Since  the  motion  of  the  describing  point  is  not  limited  to  one  com- 
plete revolution,  it  may  continue  indefinitely.  The  part  drawn  is 
one  complete  portion  and  any  addition  is  but  the  repetition  of  the 
same  moved  along  the  axial  length  of  the  curve.  The  proportions 
of  the  curve  may  vary  between  wide  limits  depending  on  the  rela- 
tive size  of  the  radius  of  the  cylinder  to  the  axial  movement  for  one 
revolution.  This  axial  distance  is  known  as  the  "pitch"  of  the 
helix. 

In  Fig.  46  the  pitch  is  about  three  times  the  radius  of  the  helix. 
In  Fig.  47,  a  short-pitch  helix  is  represented,  the  pitch  being  about 
J  the  radius,  and  a  number  of  complete  rotations  being  shown. 

The  proportions  of  the  helix  depends  therefore  on  the  radius  and 
on  the  pitch.  To  execute  a  drawing,  such  as  Fig.  46,  describe  first 
the  view  of  the  helix  which  is  a  circle.  Divide  the  circumference 
into  any  number  of  equal  parts  (12  or  24  usually).  From  these 
points  of  division  project  lines  to  the  other  view  or  views.  Divide 
the  pitch  into  the  same  number  of  equal  parts,  and  draw  lines  per- 
pendicular to  those  already  drawn.     Pass  a  smooth  curve  through 


68 


Engineering  Descriptive  Geometry 


the  points  of  intersection  of  these  lines,  forming  the  continuous 
diagonal.  In  Figs.  45  and  46  the  helix  is  a  "  right-hand  helix/' 
The  upper  part  of  Fig.  47  shows  a  left-hand  helix,  the  motion  of 
rotation  being  reversed,  or  from  12  to  11  to  10,  etc.    The  ordinary 


J-* 

2 

3 

^ 

—      ~ 

? 

i 

zTZ 

Fig.  46. 


Fig.  47. 


screw  thread  used  in  machinery  is  a  very  short-pitched  right-hand 
helix.  It  is  so  short  indeed  that  it  is  customary  to  represent  the 
curve  by  a  straight  line  passing  through  those  points  which  would 
be  given  if  the  construction  were  reduced  to  dividing  the  circum- 


Curved  Lines 


59 


ference  and  the  pitch  into  2  equal  parts.  This  is  shown  in  the 
lower  part  of  Fig.  47,  where  only  the  points  0,  6  and  12  have  been 
used. 

The  concealed  portion  of  the  helix  is  then  omitted  entirely,  no 
broken  line  for  the  hidden  part  being  allowed  by  good  practice. 

56.  The  Curved  Line  in  Space. — A  curve  in  space  may  some- 
times be  required,  one  which  follows  no  known  mathematical  law, 
but  which  passes  through  certain  points  given  by  their  coordinates. 
For  example,  in  Fig.  48,  four  points,  A    (12,1,9),  B   (5,4,6), 


Fig.  48. 


C  (2,4,4)  and  D  (2,5,1),  were  taken  as  given  and  a  "smooth 
curve/^  the  most  natural  and  easy  curve  possible,  has  been  passed 
through  them.  It  is  fairly  easy  to  pass  smooth  curves  through  the 
projections  of  the  4  points  on  each  reference  plane,  but  it  is  essen- 
tial that  not  only  should  the  original  points  obey  the  laws  of  pro- 
jection of  Art.  11,  but  every  intermediate  point  as  well.  The 
views  must  check  therefore  point  by  point  and  the  process  of  trac- 
ing the  curve  must  be  carried  out  about  as  follows:  The  projec- 
tions of  the  4  points  on  V  a^d  §  are  seen  to  be  more  evenly  ex- 
tended than  those  on  ff,  and  smooth  curves  are  made  to  pass 
through  them  by  careful  fitting  with  the  draftsman's  curves.    The 


60  Engineering  Descriptive  Geometry 

view  on  ff  cannot  now  be  put  in  at  random,  but  must  be  constructed 
to  correspond  to  the  other  views.  To  fill  in  the  wide  gap  between 
Ah  and  Bn  an  intermediate  point  is  taken,  as  Ev  on  AvBv.  By  a 
horizontal  line  Eg  is  defined.  From  Ev  and  Eg  the  ff  projection 
(Eh)  is  plotted  by  the  regular  method  of  checking  the  projections 
of  a  point.  As  many  such  intermediate  points  may  be  taken  as  may 
seem  necessary  in  each  case. 

To  define  the  sharp  turn  on  the  curve  betw'een  Cn  and  Dh,  one 
or  more  extra  points,  as  Fh,  should  be  plotted  from  the  V  and  § 
projections.  Thus  every  poorly  defined  part  is  made  definite  and 
the  views  of  the  line  mutually  check.  The  work  of  "  la3'ing  out " 
the  lines  of  a  ship  on  the  "  mold-loft  floor  "  of  a  shipbuilding  plant 
is  of  this  kind,  with  the  exception  that  the  curves  are  chiefly  plane 
curves,  not  curves  in  space. 

Problems  V. 
(For  blackboard  or  cross-section  paper.) 

50.  Make  the  descriptive  drawing  of  a  circle  lying  in  a  plane 
parallel  to  S,  center  at  0  (3,  6,  7)  and  radius  5. 

51.  Make  the  descriptive  drawing  of  a  circle  lying  in  a  plane 
perpendicular  to  V  and  making  an  angle  of  45°  with  H  (the  trace 
in  V  passing  through  the  points  (18,  0,  0),  and  (0,  0, 18)).  The 
center  of  the  circle  is  at  C  (9,  G,  9),  and  the  radius  is  5.  (Make 
the  V  projection  first,  then  a,  projection  on  an  auxiliary  plane  \]. 
From  these  views  construct  the  ff  and  S  projections,  using  8  or  9 
points. 

52.  Make  the  descriptive  drawing  of  a  circle  in  a  plane  perpen- 
dicular to  H?  the  trace  in  ff  passing  through  the  points  (12,  0,  0) 
and  (0,16,0).  The  center  is  at  (6,8,10)  and  the  radius  is  8. 
(Draw  the  plan  and  an  auxiliary  view  showing  true  shape  first,  and 
from  those  views  construct  the  projections  on  V  and  §.) 

53.  An  ellipse  lies  in  a  plane  passing  through  the  axis  of  Y  and 
making  angles  of  45"  with  ff  and  g.  The  ff  projection  is  a  circle, 
center  at  (10, 10,  0)  and  radius  8.  Prove  that  the  S  projection  is 
also  a  circle  and  find  the  true  shape  of  the  ellipse  by  revolving  the 
plane  of  the  ellipse  into  the  plane  ff. 


Curved  Lines  61 

54.  An  ellipse  lies  in  a  plane  passing  through  the  axis  of  Y  and 
making  an  angle  of  60°  with  H  and  30°  with  S-  The  H  projec- 
tion is  a  circle,  center  at  (8,  8,  0),  radius  6.  Find  the  true  shape 
of  the  ellipse.  Construct  the  view  on  §  by  projecting  points  for  the 
center  and  for  the  extremities  of  the  axes  of  the  ellipse.  Pass  a 
draftsman's  ellipse  through  those  points.  Show  that  no  appreci- 
able error  can  be  observed. 

55.  Construct  a  draftsman's  ellipse,  on  accurate  cross-section  or 
coordinate  paper,  with  major  axis  24  units,  and  minor  axis  12  units. 
Perform  the  accurate  plotting  of  the  true  ellipse  on  the  same  axes, 
one  quadrant  by  the  method  of  Fig.  42  and  one  by  the  method  of 
Fig.  43,  using  6  divisions  for  DE  and  OD.  Note  the  degree  of 
accuracy  of  the  approximate  process. 

56.  On  coordinate  paper,  plot  an  ellipse  by  the  method  of  Fig. 
43,  the  major  axis  being  16  units  long  and  the  minor  axis  8  units. 
Plot  another  ellipse  whose  major  axis  is  16  and  whose  minor  axis 
is  12.  (To  divide  the  semi-minor  axis  of  6  units  into  4  equal  parts, 
use  points  of  division  on  the  vertical  line  CE  instead  of  OD.  CE 
being  twice  as  far  from  A  as  OD,  12  units  must  be  used  for  the 
whole  length,  and  these  divided  into  4  parts.) 

57.  On  isometric  paper  pick  out  a  rhombus  like  the  top  of  Fig. 
19,  but  having  8  units  on  each  side.  Inscribe  an  ellipse  by  plotting 
by  the  method  of  Fig.  44. 

58.  Make  the  descriptive  drawing  of  a  helix  whose  axis  is  per- 
pendicular to  S  through  the  point  (0,7,7).  The  pitch  of  the 
helix  is  12,  and  the  initial  point  is  (2,  7,  2).  Draw  the  ff  and  V 
projections  of  a  right-hand  helix,  numbering  the  points  in  logical 
order. 

59.  Connect  the  4  points  A  (10,  8, 10),  B  (8, 10,  6),  C  (6,  9,  4) 
and  D  (2,  2,  4)  by  a  smooth  curve,  filling  out  poorly  defined  por- 
tions in  S  from  the  ff  and  V  projections. 


CHAPTER  VI. 
CTTRVED  SURFACES  AND  THEIR  ELEMENTS. 

57.  Lines  Representing  Curved  Surfaces. — To  represent  solids 
having  curved  surfaces,  it  is  not  enough  to  represent  the  actual 
comers  or  edges  only.  Hitherto  only  edges  have  appeared  on  de- 
scriptive drawings,  and  it  has  been  a  feature  of  the  drawings  that 
every  point  represented  on  one  projection  must  be  represented  on 
the  other  projections,  the  relation  between  projections  being  strictly 
according  to  rule.  We  now  come  to  a  class  of  lines  which  do  not 
appear  on  all  three  views,  lines  due  to  the  curvature  of  the  surfaces. 

The  general  principle,  called  the  "  Principle  of  Tangent  Projec- 
tors," governing  this  new  class  of  lines  is  as  follows:  In  projecting 
a  curved  surface  to  a  given  plane  of  projection  (by  perpendicular 
projectors,  of  course)  all  points,  and  only  those  points,  whose  pro- 
jectors are  tangent  to  the  curved  surface  should  be  projected.  A 
good  illustration  of  this  principle  is  shown  in  Fig,  45,  where  the 
cylinder  is  projected  upon  the  plane  of  projection.  The  top  and 
bottom  bases  are  edges,  and  project  under  the  ordinary  rules,  but 
along  the  straight  line  0, 1,  2,  . . .  .,  12  the  curved  surface  of  the 
cylinder  is  itself  perpendicular  to  the  plane  of  projection.  If  from 
any  point  on  this  line  a  projector  is  drawn  to  the  plane  of  projection 
(as  is  shown  in  the  figure  for  the  points  1,  2,  3,  etc.),  this  projector 
is  tangent  to  the  cylinder.  The  whole  line  therefore  projects  to 
the  plane  of  projection.  The  projection  of  the  cylinder  on  a  plane 
parallel  to  its  axis  is  therefore  a  rectangle,  two  of  its  sides  repre- 
senting the  circular  bases  and  the  two  other  edges  representing  the 
curved  sides  of  the  cylinder. 

58.  The  Right  Circular  Cylinder. — The  complete  descriptive 
drawing  of  a  cylinder  is  therefore  as  shown  in  Fig.  49.  This  cylin- 
der is  a  right  circular  cylinder.  Mathematicians  consider  that  the 
cylinder  is  "  generated  "  by  revolving  the  line  A  A'  about  PP'y  the 
axis  of  the  cylinder.     The  generating  line  in  any  particular  posi- 


Curved  Surfaces  and  Their  Elements 


63 


tion  is  called  an  "  element ''  of  the  surface.  Thus  AA',  BB',  CC, 
etc.,  are  elements. 

When  the  cylinder  is  projected  upon  V,  AA'  and  CC  are  the 
elements  which  appear  in  V  because  the  V  projectors  of  all  points 
along  those  lines  are  tangent  to  the  cylinder,  as  can  be  seen  from 
the  view  on  H-  The  elements  which  are  represented  by  lines  on 
S  are  BB'  and  DD'. 

The  right  circular  cylinder  may  also  be  considered  as  generated 
by  moving  a  circle  along  an  axis  perpendicular  to  its  own  plane 
through  its  center. 


If 

D 

P    ■ 

4 

S? 

B 

■  X 

P        0 

r     ^ 

A 

DB 

t 

b 

CA 

D 

V 

s 

A 

f'j 

7. 

B 

JD' 

4 

B' 

c'l^ 

D' 

B^^B 


Fig.  49. 


Fig.  50. 


In  Fig.  45  consider  the  top  base  of  the  cylinder  to  be  moved 
down  the  cylinder.  Each  successive  position  of  the  circle  is  a  "  cir- 
cular element"  of  the  cylinder.  The  circles  through  the  points 
1,  2,  3,  etc.,  are  simply  circular  elements  of  the  cylinder  taken  at 
equal  distances  apart. 

59.  The  Inclined  Circular  Cylinder. — Fig.  50  shows  an  inclined 
circular  cylinder.  It  has  circular  and  straight  line  elements  as 
before,  though  it  cannot  be  generated  by  revolving  a  line  about 
another  at  a  fixed  distance,  but  can  be  generated  by  moving  the 
circle  ABCD  obliquely  to  A'B'C'D',  the  center  moving  on  the  axis 
PP\  The  straight  elements  are  all  parallel  to  the  axis.  The  cross- 
section  of  a  cylinder  is  a  section  taken  perpendicular  to  the  axis. 


64 


Engineering  Descriptive  Geometry 


In  this  case  the  cross-section  is  an  ellipse,  and  for  this  reason  the 
Inclined  Circular  Cylinder  is  sometimes  called  the  Elliptical  Cyl- 
inder. 

60.  Straight  and  Inclined  Circular  Cones. — If  a  generating  line 
AP,  Fig.  51,  meets  an  axis  PP'  at  a  point  P,  and  is  revolved  about 
it,  it  will  generate  a  Straight  Circular  Cone.  The  cone  has  both 
straight  and  circular  elements,  the  circular  elements  increasing  in 
size  as  they  recede  from  the  vertex  P.  The  base  A  BCD  is  one  of 
the  elements. 


The  Inclined  Circular  Cone  (Fig.  52)  has  straight  and  circular 
elements,  but  it  is  not  generated  by  revolving  a  line  about  the  axis. 
The  circular  elements  move  obliquely  along  the  axis  PP'  and  in- 
crease uniformly  as  they  recede  from  the  vertex  P. 

61.  The  Sphere. — ^The  Sphere  can  be  generated  by  revolving  a 
semiciicle  about  a  diameter.  Each  point  generates  a  circle,  the 
radii  of  the  circles  for  successive  points  having  values  varying 
between  0  and  the  radius  of  the  sphere.  Since  the  sphere  can  be 
generated  by  using  any  diameter  as  an  axis,  the  number  of  ways  in 
which  the  surface  can  be  divided  into  circular  elements  is  infinite. 

62.  Surfaces  of  Revolution. — In  general,  any  line,  straight  or 
curved,  may  be  revolved  about  an  axis,  thus  creating  a  surface  of 
revolution.    Every  point  on  the  "  generating  line  "  creates  a  "  cir- 


CuKVED  Surfaces  and  Their  Elements 


65 


cular  element "  of  the  surface,  and  the  plane  of  each  circular  ele- 
ment is  perpendicular  to  the  axis  of  the  surface. 

The  straight  circular  cylinder  is  a  simple  case  of  the  general 
class  of  surfaces  of  revolution.  To  generate  it  a  straight  line  is 
revolved  about  a  parallel  straight  line.  The  different  points  of  the 
generating  line  create  the  circular  elements  of  the  cylinder,  and 


'M^' 


; 


Fig.  53. 


Fig.  54. 


the  different  positions  of  the  generating  line  mark  the  straight  ele- 
ments. The  cone  and  the  sphere  are  also  surfaces  of  revolution,  as 
they  are  generated  by  revolving  a  line  about  an  axis. 

If  a  circle  be  revolved  about  an  axis  in  its  own  plane,  but  en- 
tirely exterior  to  the  circle,  a  solid,  called  an  "  anchor  ring,"  is 
generated.  A  small  portion  of  this  surface,  part  of  its  inner  surface, 
is  often  spoken  of  as  a  "  bell-shaped  surface,'*  from  its  similarity 
to  the  flaring  edge  of  a  bell. 

Any  curved  line  may  create  a  surface  of  revolution,  but  in  de- 


66  Engineering  Descriptive  Geometry 

signs  of  machinery  lines  made  np  of  parts  of  circles  and  straight 
lines  are  most  frequently  used.  Figs.  53  and  54  show  two  exam- 
ples which  illustrate  well  the  application  of  the  Principle  of  Tan- 
gent Projectors.  The  generating  line  is  emphasized  and  the  cen- 
ters of  the  various  arcs  are  marked. 

Any  angular  point  on  the  generating  line,  as  a  (Fig.  53),  creates 
a  circular  edge  on  the  surface.  This  edge  appears  as  a  circle  on  the 
plan  (as  aa'  on  li),  and  as  a  straight  line,  equal  to  the  diameter, 
on  the  elevation  (as  aa'  on  V)-  See  also  the  point  h  (Fig.  54). 
In  addition,  any  portion  of  the  generating  line  which  is  perpen- 
dicular to  the  axis,  as  h  (Fig.  53),  even  if  for  an  infinitely  short 
distance  only,  creates  a  line  on  the  side  view,  as  hh'  on  V?  but  no 
corresponding  circle  on  H.  A  V  projector  from  any  point  on  the 
circular  element  created  by  the  point  h  is  tangent  to  the  surface, 
and  therefore  creates  a  point  on  the  drawing,  but  an  f\  projector 
is  not  tangent  to  the  surface,  e  is  a  similar  point,  and  so  also  is 
j  of  Fig.  54. 

Any  point,  as  c,  Fig.  53,  where  the  generating  line  is  parallel  to 
ihe  axis  for  a  finite,  or  for  an  infinitely  small  distance,  generates 
a  circular  element,  from  every  point  of  which  the  J-f  projectors  are 
tangent  to  the  surface,  but  the  V  projectors  are  not.  A  circle  c(f 
appears,  therefore,  on  the  plan  for  this  element  of  the  surface  of 
revolution,  but  no  straight  line  on  the  side  view.  J  is  a  similar 
point,  as  are  also  /  and  g,  on  Fig.  54. 

63.  The  Helicoidal  Surface. — If  a  line,  straight  or  curved,  is 
made  to  revolve  uniformly  about  an  axis  and  move  uniformly  along 
the  axis  at  the  same  time,  every  point  in  the  line  will  generate  a 
helix  of  the  same  pitch.  The  surface  swept  up  is  called  a  Heli- 
coidal Surface. 

The  generating  line  chosen  is  usually  a  straight  line  intersecting 
the  axis.  The  surfaces  used  for  screw  threads  are  nearly  all  of 
this  kind.  Fig.  55  gives  an  example  of  a  sharp  V-threaded  screw, 
the  two  surfaces  of  the  thread  having  been  generated  by  lines  in- 
clined at  an  angle  of  60°  to  the  axis.  Fig.  56  shows  a  square 
thread,  the  generating  lines  of  the  two  helicoidal  surfaces  being 
perpendicular  to  the  axis.  Any  particular  position  of  the  straight 
line  is  a  "  straight  element "  of  the  helicoidal  surface. 


Curved  Surfaces  and  Their  Elements 


67 


64.  Elementary  Intersections. — In  executing  drawings  of  ma- 
cliinery  it  is  often  necessary  to  determine  the  line  of  intersection  of 
two  surfaces,  plane  or  curved.  The  simplest  lines  of  intersection 
are  such  as  coincide  with  elements  of  a  curved  surface.    They  may 


Fig.  55. 


Fig.  56. 


be  called  "  Elementary  Intersections."  An  elementary  intersection 
may  arise  when  a  curved  surface  is  intersected  by  a  plane,  so  placed 
as  to  bear  some  simple  relation  to  the  surface  itself. 

In  Fig.  49,  any  plane  perpendicular  to  the  axis  of  the  cylinder 
intersects  it  in  a  circular  element  of  the  cylinder,  and  any  plane 
parallel  to  the  axis  of  the  cylinder   (or  containing  it)   intersects 


68  Engineering  Desceipttve  Geometry 

it  (if  it  intersects  it  at  all)  in  two  straight  line  elements  of  the 
cylinder. 

In  Fig.  50  any  plane  parallel  to  the  base  of  the  cylinder  inter- 
sects it  in  a  circular  element,  and  any  plane  parallel  to  the  axis, 
or  containing  it,  intersects  it  in  straight  elements  of  the  cylinder. 

In  Fig.  51  or  52  any  plane  parallel  to  the  base  of  the  cone  inter- 
sects it  in  a  circular  element,  and  any  plane  containing  the  vertex 
of  the  cone  (if  it  intersects  at  all)  intersects  the  cone  in  straight 
elements. 

In  Fig.  53  or  54  any  plane  perpendicular  to  the  axis  of  the  sur- 
face of  revolution  intersects  it  in  a  circular  element. 

In  Fig.  55  or  56  any  plane  containing  the  axis  of  the  screw  inter- 
sects the  helicoidal  surfaces  in  straight  elements.  The  plane  per- 
pendicular to  ff,  cutting  H  in  a  trace  PQ,  and  cutting  V  in  a 
trace  QR,  cuts  the  helicoidal  surfaces  at  each  convolution  in  straight 
elements.     Only  ah  and  a'h  are  marked  on  the  figure. 

Problems  VI. 

(For  blackboard  or  cross-section  paper  or  wire-mesh  cage.) 

60.  Draw  the  projections  of  a  cylinder  whose  axis  is  P  (6,  2,  6), 
P'  (6, 16,  6),  and  radius  5.  Draw  the  intersection  of  this  cylinder 
with  a  plane  parallel  to  fi,  at  4  units  from  H,  and  with  a  plane 
parallel  to  V?  10  units  from  V- 

61.  An  inclined  circular  cylinder  has  its  bases  parallel  to  S-  Its 
axis  is  P  (2,  7,  7),  P'  (14,  7, 13).  Its  radius  is  5.  Draw  the  V 
and  S  projections  and  the  intersection,  with  a  plane  parallel  to  §, 
6  units  from  S,  and  with  a  plane  parallel  to  V?  3  units  from  V» 

62.  Draw  a  cone  with  vertex  at  P  (4,8,8),  center  of  base  at 
P'  (16,  8,  8),  and  radius  6,  the  base  lying  in  a  plane  parallel  to  S- 
Draw  the  intersection  with  a  plane  parallel  to  §,  12  units  from  S, 
and  with  a  plane  perpendicular  to  §,  whose  trace  in  S  passes 
through  the  points  (0,  8,  8)  and  (0, 14,  0). 

63.  An  oblique  cone  has  its  vertex  at  P  (16,  8,  4)  and  its  base  in 
a  plane  parallel  to  ff,  center  at  P'  (8,  8, 16),  and  radius  5.  Draw 
the  intersection  with  a  plane  parallel  to  H?  13  units  from  H,  and 
with  a  plane  containing  the  axis  and  the  point  (16,  0, 16). 


Curved  Surfaces  and  Their  Elements  69 

G4.  A  cone  has  an  axis  P  (8,  2,  2),  F  (8, 14, 10).  Its  base  is  in 
a  plane  parallel  to  V?  l-i  units  from  \ ,  and  its  radius  is  6  units. 
Draw  the  intersection  with  a  plane  containing  the  vertex  and  the 
points  (0,14,12)  and  (16,14,12). 

65.  A  surface  of  revolution  is  formed  by  revolving  a  circle,  whose 
center  is  at  (12,  8,  8)  and  radius  3  units,  lying  in  a  plane  parallel 
to  Vj  about  an  axis  perpendicular  to  W  at  the  point  (8,  8,  0).  It 
is  cut  by  a  plane  parallel  to  H  at  a  distance  of  6  units  from  f\. 
Draw  the  intersections. 

QQ.  A  sphere  has  its  center  at  (8,  8,  8)  and  a  radius  of  5  units. 
Draw  the  intersection  with  a  cylinder  whose  axis  is  P  (8,8,0), 
P'  (8,8,16),  and  whose  radius  is  4  units,  its  bases  being  planes 
perpendicular  to  its  axis. 

67.  A  sphere  has  its  center  at  (8,  8,  8)  and  a  radius  of  5  units. 
Find  its  intersection  with  a  cone  whose  vertex  is  P  (0,  8,  8),  center 
of  base  (16,  8,  8),  and  radius  of  base  6  units,  the  base  being  in  a 
plane  §'  parallel  to  S- 

68.  In  Fig.  53  let  the  generating  line  Pahcde  be  revolved  about 
ee'  as  an  axis.     Assume  any  dimensions  for  the  line  and  draw  the 

V  and  §  projections  of  the  surface  of  revolution  thus  formed. 
Draw  tbe  intersection  with  a  plane  parallel  to  S  just  to  the  right 
of  d. 

69.  In  Fig.  54  let  the  generating  line  Pfgh  be  revolved  about 
liW  as  an  axis.    Assume  any  dimensions  for  the  line  and  draw  the 

V  and  §  projections  of  the  surface  of  revolution  formed. 


CHAPTEE  VII. 
INTERSECTIONS  OF  CURVED  SURFACES. 

65.  The  Method  of  the  Intersection  of  the  Intersections. — The 

determination  of  the  line  of  intersection  of  two  curved  surfaces  (or 
of  a  curved  surface  and  a  plane),  when  not  an  "  Elementary  Inter- 
section," is  of  much  greater  difficulty  and  requires  a  clear  under- 
standing of  the  nature  of  the  curved  surfaces  themselves,  and  some 
little  ingenuity  in  applying  general  principles. 

The  method  generally  relied  upon  for  the  solution  is  the  use  of 
auxiliary  intersecting  planes  so  chosen  as  to  cut  elementary  inter- 
sections with  each  of  the  given  surfaces.  These  elementary  inter- 
sections are  drawn  and  the  points  of  intersection  of  the  intersec- 
tions are  identified  and  recorded  as  points  on  the  required  line  of 
intersection.  This  method  is  spoken  of  as  "  finding  the  intersec- 
tion of  the  intersections.^'  When  a  number  of  auxiliary  planes 
have  been  used  in  this  way,  a  smooth  curve  is  passed  through  the 
points  on  the  required  intersection  of  the  surfaces,  as  described  in 
Art.  55.  It  should  not  be  necessary,  however,  to  interpolate  points 
to  fill  out  gaps  as  was  done  in  Fig.  48  for  E  and  F.  This  can  be 
done  better  by  the  use  of  more  auxiliary  intersecting  planes.  Ex- 
amples of  tliis  method  will  make  it  clear. 

66.  An  Inclined  Circular  Cylinder  Cut  by  an  Inclined  Plane. — 
In  Fig,  57  an  inclined  cylinder,  axis  PP',  is  cut  by  a  plane  perpen- 
dicular to  Y>  and  inclined  to  ff.  The  traces  of  this  plane  are  IJ 
in  H,  JK  in  V,  and  KL  in  S- 

It  is  an  Inclined  Plane  (see  Art.  19),  not  an  Oblique  Plane. 
Having  the  descriptive  drawing  of  the  cylinder  and  the  traces  of 
the  plane  given,  the  problem  is  to  draw  the  line  of  intersection  of 
the  surfaces.  It  is  well-known  that  in  this  case  the  line  of  inter- 
section is  an  ellipse,  but  the  method  of  determining  it  permits  the 
ellipse  to  be  plotted  whether  it  is  recognized  as  such  or  not.  No 
use  is  to  be  made  of  previous  knowledge  of  the  nature  of  the  curve 


Intersections  of  Curved  Surfaces 


71 


of  intersection  of  any  of  the  cases  treated  in  this  and  the  next 
chapter. 

Two  variations  of  the  method  are  applicable  in  this  case.  In  the 
first  method,  auxiliary  intersecting  planes  may  be  taken  parallel  to 
the  axis  of  the  cylinder.  The  simplest  method  of  doing  this  is  to 
take  auxiliary  planes  parallel  to  V?  since  the  axis  itself  is  parallel 


Fig.  57. 


to  V.    Let  R'R  be  the  trace  on  H,  and  RR"  the  trace  on  S  of  a 
plane  parallel  to  V-    We  may  call  this  plane  simply  "'  R.'' 

Let  e  and  /  be  the  points  where  R'R  cuts  the  top  base  of  the  cyl- 
inder. Project  these  points  from  Hi  to  V  and  in  V  draw  ee'  and 
ff  parallel  to  PP\  These  straight  elements  of  the  cylinder  are  the 
lines  of  intersection  of  the  auxiliary  plane  with  the  cylinder.  As 
a  check  on  the  work,  e'  and  f ,  where  R'R  in  H  cuts  the  bottom 
base  of  the  cylinder,  should  project  vertically  to  e'  and  f  in  V- 


7^  Engineering  Descriptive  Geometry 

The  auxiliary  plane  cuts  the  given  plane  JK  in  a  line  of  inter- 
section whose  projection  on  V  coincides  with  JK  itself. 

The  points  ;'  and  h,  where  ee'  and  ff  intersect  JK,  are  the  ""  inter- 
sections of  the  intersections/'  and  are  therefore  points  on  the  line 
cf  intersection  of  the  cylinder  and  the  plane  K.  Project  j  and  h 
to  R'R  on  H  and  to  RR"  on  S.  These  are  points  on  the  required 
curves  in  ff  and  g.  By  extending  in  H  the  projecting  lines  of 
j  and  Ic  as  far  above  the  axis  PP'  as  ;  and  h  are  below  it,  ;'  and  k', 
points  on  the  upper  half  of  the  cylinder,  symmetrical  with  ;  and  k 
en  the  lower  half,  are  found.  The  construction  is  equivalent  to 
passing  a  second  auxiliary  plane  parallel  to  PP^  at  the  same  dis- 
tance from  PP'  as  R,  but  on  the  other  side. 

By  passing  a  number  of  planes  similar  to  R,  a  sufficient  number 
of  points  are  located  to  define  accurately  the  ellipse  ahcd  in  H 

andS. 

The  true  shape  of  this  ellipse  is  shown  in  JJ?  a  plane  parallel  to 
JK,  at  any  convenient  distance.  In  the  example  chosen,  the  plane 
JK  has  been  taken  perpendicular  to  PP\  so  that  the  ellipse  abed  is 
the  true  cross-section  of  the  cylinder.  Nothing  in  the  method  de- 
pends on  this  fact  and  it  is  perfectly  general  and  applicable  to  any 
inclined  plane. 

A  variation  may  be  made  by  passing  the  auxiliary  planes  per- 
pendicular to  V  and  parallel  to  PP\  ee'  in  V  J^^ay  be  taken  as  the 
trace  of  such  a  plane.  The  intersections  of  this  auxiliary  with  both 
surfaces  should  be  traced  and  the  intersection  of  the  intersections 
identified  and  recorded  as  a  point  of  the  curve  required.  /  and  / 
are  the  points  thus  found.  This  method  indeed  requires  the  same 
construction  lines  as  before,  but  gives  a  different  explanation  to 
them. 

67.  A  Second  Method  Using  Circular  Elements  of  the  Cylinder. — 
A  plane  parallel  to  the  base  of  the  cylinder  and  therefore,  in  this 
case,  parallel  to  H,  will  cut  the  cylinder  in  a  line  of  intersection 
which  is  one  of  the  circular  elements  of  the  cylinder.  Let  T'T  and 
TT",  in  Fig.  58,  be  the  traces  of  a  plane  "  T"  parallel  to  H-  The 
axis  of  the  cylinder  PP'  pierces  the  plane  T  at  p.  p  is  therefore 
tl:e  center  of  the  circle  of  intersection  of  the  auxiliary  plane  T  with 
the  cylinder.  Project  p  to  ff,  and  using  ,p  as  a  center  and  with  a 
radius  equal  to  pt,  describe  the  circle  as  shown. 


Intersections  of  Curved  Surfaces 


73 


The  planes  T  and  JK  are  both,  perpendicular  to  V  or  "  seen  on 
edge  "  in  V-  Their  line  of  intersection  is  therefore  perpendicular 
to  V>  or  is  "  seen  on  end  "  in  V?  as  the  point  j.  Project  /to  H? 
where  it  appears  as  the  line  ;;'.  This  line  is  the  intersection  of  the 
two  planes. 

The  points  j  and  ;',  where  this  line  of  intersection  ;/  meets  the 
circular  intersection  whose  center  is  at  p,  are  the  "  intersections  of 
the  intersections/'  and  are  points  on  the  required  curve. 

N  I 


Fig.  58. 


Planes  like  T,  at  various  heights  on  the  cylinder,  determine  pairs 
of  points  on  the  curve  of  intersection  on  H-  From  H  and  V  the 
points  may  be  plotted  on  S  by  the  usual  rules  of  projection,  thus 
completing  the  solution. 

68.  Singular  or  Critical  Points. — It  is  nearly  always  found  that 
one  or  two  points  on  the  line  of  intersection  may  be  projected  di- 
rectly from  some  one  view  to  the  others  without  new  construction 
lines.    In  this  case  a  and  c  in  \,  Fig.  57,  may  be  projected  at  once 


74 


Engineering  Descriptive  Geometry 


to  IM  and  S.  They  correspond  theoretically  to  points  determined 
by  a  central  plane,  cutting  fi  in  a  trace  PP'.  h  and  d  may  also  be 
projected  directly,  as  they  correspond  to  planes  whose  traces  in 
IHI  are  BB'  and  DD\  These  critical  points  sliould  always  be  the 
first  points  identified  and  recorded,  though  usually  no  explanation 
will  be  given,  as  they  should  be  obvious  to  any  one  who  has  grasped 
the  general  method. 

69.  A  Cone  Intersected  by  an  Inclined  Plane. — Fig.  59  shows 


the  descriptive  drawing  of  a  right  circular  cone  intersected  by  an 
inclined  plane  whose  traces  are  JK  and  KL,  Two  methods  of  solu- 
tion are  shown. 

A  plane  R,  containing  the  axis  PP',  and  therefore  perpendicular 
to  fi,  is  shown  by  its  traces  R'R  and  RR".  It  intersects  the  cone 
in  the  elements  Pj  and  Ph.  From  JHI  project  these  points  /  and  Ic 
to  Vj  and  draw  the  elements  in  V-  The  V  projection  of  the  inter- 
section of  R  with  the  plane  JK  is  the  line  JK,  and  the  points  e  and 
/  are  the  intersections  of  the  intersections,     e  and  /  are  now  pro- 


Intersections  of  Curved  Surfaces 


75 


jected  to  the  plan,  where  they  necessarily  lie  on  the  line  BE".  Sym- 
metrical points  e'  and  f  are  also  plotted  and  all  four  points  trans- 
ferred to  the  side  elevation. 

A  plane  T  perpendicular  to  the  axis  PP'  whose  traces  are  T'T 
and  TT"  may  be  used  instead  of  R.  Its  intersection  with  the  cone 
is  a  circle,  seen  on  edge  in  the  front  elevation  as  the  line  hh'.  Its 
center  is  g,  and  radius  is  gh.     Draw  this  circle  in  the  plan.     The 


intersection  of  T  with  the  plane  JK  is  a  line,  seen  on  end,  as  the 
point  /  of  the  front  elevation.  Draw  /'/  in  IHI  as  this  line.  The 
points  /  and  /'  are  the  intersections  of  the  intersections. 

70.  Intersection  of  Two  Cylinders. — Fig.  60  shows  the  inter- 
section of  two  cylinders.  Since  they  are  right  cylinders,  and  their 
axes  are  at  right  angles,  planes  parallel  to  any  one  of  the  three 
reference  planes  will  cut  only  straight  or  circular  elements  of  the 
cylinders.    By  the  solution.  Fig.  60,  auxiliary  planes  parallel  to  V 


76 


Engineerixg  Descriptive  Geometry 


liave  been  chosen,  the  traces  of  one  being  R'R  and  BE".  This  plane 
intersects  the  vertical  cylinder  in  the  lines  IcTc'  and  IV,  and,  it  inter- 
sects the  horizontal  cylinder  in  the  lines  mm!  and  nn'.  The  inter- 
sections of  these  intersections  are  the  points  marked  r. 

If  the  axes  of  the  cylinders  do  not  meet  but  pass  at  right  angles, 
no  new  complication  is  introduced.  If  the  axes  of  the  cylinders 
meet  at  an  angle,  and  one  or  both  cylinders  are  inclined,  the  choice 


of  methods  may  be  greatly  reduced,  but  one  method  is  always  pos- 
sible. To  discover  it,  try  planes  parallel  to  the  axes  of  both  cylin- 
ders, or  parallel  to  one  axis  and  to  one  plane  of  reference;  or  in 
some  manner  bearing  a  definite  relation  to  the  nature  of  the  sur- 
faces. 

71.  Intersection  of  a  Cylinder  and  a  Sphere. — In  Fig.  61  a 
sphere  is  intersected  by  a  cylinder,  whose  axis  PP'  does  not  pass 
through  the  center  of  the  sphere  at  §.     In  the  solution.  Fig.  61, 


Intersections  of  Curved  Surfaces 


77 


auxiliary  planes  parallel  to  V  have  been  chosen,  the  traces  of  one  of 
them  being  E'E  and  EE".  The  plane  E  cuts  the  sphere  in  a  circle 
whose  diameter  is  eg,  as  given  by  the  plan.  This  circle  is  described 
in  V-  The  intersections  of  this  circle  with  the  elements  of  the 
cylinder  W  and  IV  are  the  points  marked  r,  points  on  the  required 
curve  of  intersection. 

In  this  case  the  points  are  first  determined  on  the  front  elevation 
and  then  projected  to  the  side  elevation.  Solutions  by  planes  par- 
allel to  ffi  or  to  S  may  be  made,  requiring  however  different  con- 
struction lines. 


Pig.  62. 


72.  Intersection  of  a  Cone  and  a  Cylinder:  Axes  Intersecting. — 

In  Fig.  62  a  cone  and  a  cylinder  intersect  at  right  angles.     The 
solution  chosen  is  by  horizontal  planes,  as  T. 

An  alternate  solution  is  by  planes  perpendicular  to  S,  and  con- 
taining the  point  P.  The  planes  must  cut  both  surfaces,  and  their 
traces,  where  seen  on  edge,  as  PE,  Fig.  62,  must  cut  the  projections 
of  both  surfaces.  These  two  solutions  hold  good  even  if  the  axes 
do  not  meet  but  pass  each  other  at  right  angles. 


.78 


Engineering  Descriptive  Geometry 


If  the  axes  are  not  at  right  angles,  modifications  must  be  made, 
and  the  search  for  a  system  of  planes  making  elementary  intersec- 
tions with  hotli  surfaces  requires  some  ingenuity  and  thought. 

73.  Intersection  of  a  Cone  and  Cylinder :  Axes  Parallel. — A 
simple  case  is  shown  in  Fig.  63.  Two  methods  of  solution  are  avail- 
able.    In  one,  horizontal  planes  are  used.     Each  plane,  such  as  T, 


Fig.  63. 


makes  circular  intersections,  with  both  cone  and  cylinder,  the  inter- 
sections intersecting  at  points  t  and  t.  A  second  method  is  by 
planes  perpendicular  to  H,  containing  the  axis  PP'.  One  plane 
''  R  "  is  shown  by  its  traces  WP  in  H  and  PP"  in  S.  this  plane 
being  taken  so  as  to  give  the  same  point  t  on  the  curve  and  another 
point  t' .  In  the  execution  of  drawings  of  this  class  it  is  natural  to 
take  the  auxiliary  planes  at  regular  intervals  if  the  planes  are 
parallel  to  each  other,  or  at  equal  angles  if  the  planes  radiate  from 
a  central  axis. 


Inteksections  of  Curved  Surfaces  79 

Problems  VII. 

'  70.  An  inclined  cylinder  has  one  base  in  H  and  one  in  a  plane 
parallel  to  H-  Its  axis  is  P  (11,  8,  0),  F  (5,  8, 16) .  Its  radius  is 
4  units.  It  is  intersected  by  a  plane  perpendicular  to  Yy  whose 
trace  passes  through  the  points  (5,  0,  0)  and  (11,  0, 16).  Draw  the 
three  projections  and  show  one  intersecting  auxiliary  plane  by  con- 
struction lines. 

71.  A  cone  has  its  vertex  in  H  at  (6,6,0)  and  its  base  in  a 
plane  parallel  to  IM,  center  at  (6,  6, 12),  and  radius  5.  It  is  inter- 
sected by  a  plane  containing  the  axis  of  Y  and  making  angles  of 
45°  with  H  and  S.    Draw  the  projections. 

72.  A  cone  has  its  vertex  at  (2, 14, 16)  and  its  base  is  a  circle 
in  flf  center  at  (8,  8,  0),  and  radius  6.  Find  its  intersection  with 
a  vertical  plane  4  units  from  g. 

73.  A  right  circular  cylinder  has  its  base  in  S,  center  at  (0,  8,  8), 
and  radius  4.  Its  axis  is  16  units  long.  Another  right  cylinder 
has  its  base  in  H,  center  at  (8,  8,0),  radius  5,  and  axis  16  units 
long.  Draw  their  lines  of  intersection,  the  smaller  cylinder  being 
supposed  to  pierce  the  larger. 

74.  A  right  circular  cylinder  has  its  base  in  S,  center  at  (0,  7,  8), 
and  radius  4.  Its  axis  is  16  units  long.  Another  right  circular 
cylinder  has  its  base  in  fl,  center  at  (8,  9,  0),  radius  5,  and  axis  16 
units  long.  Draw  their  line  of  intersection,  the  smaller  cylinder 
being  supposed  to  pierce  the  larger. 

75.  Two  inclined  circular  cylinders  of  3  units  radius  have  their 
bases  in  H  and  in  fj'  (16  units  from  ff).  The  axis  of  one  is 
P  (4,8,0),  P'  (12,8,16),  and  of  the  other  is  Q  (12,8,0), 
Q'  (4,8,16).  Prove  that  their  intersection  consists  of  two  parts, 
one  a  circle  in  a  plane  parallel  to  H,  and  one  an  ellipse  in  a  plane 
parallel  to  §. 

76.  A  sphere  has  its  center  at  (8,  9,8),  and  radius  6^  units.  A 
vertical  right  circular  cylinder  has  its  top  base  in  ff,  center  at 
(8,6,0),  radius  4,  and  length  16  units.  Find  the  intersection  of 
the  surfaces. 

77.  A  right  circular  cylinder,  axis  P  (0,8,9),  P'  (16,8,9), 
radius  5,  is  pierced  by  a  right  circular  cone.    The  base  of  the  cone 


80  Engineering  Descriptive  Geometry 

is  in  a  plane  16  iinits  from  ff,  center  at  Q'  (8,  8,  16),  and  radius 
6.  The  vertex  of  the  cone  is  at  §  (8,  8,  0).  Find  the  lines  of  inter- 
section. 

78.  An  inclined  cylinder  has  an  oblique  line  P  (0,11,5), 
P'  (16,5,11)  for  its  axis.  The  radius  of  the  circular  base  is  4 
units  and  the  planes  of  the  bases  are  S,  and  S'  parallel  to  S  at  16 
units'  distance.  The  cylinder  is  cut  by  a  plane  parallel  to  V  at  7 
units'  distance  from  V-  Draw  the  three  projections  of  the  cylinder 
and  the  line  of  intersection. 

79.  An  inclined  cylinder  has  an  oblique  line  P,(0, 11, 5), 
P'  (16,  5, 11)  for  its  axis.  The  radius  of  the  circular  base  is  4 
units,  and  the  planes  of  the  bases  are  §,  and  §'  parallel  to  S  at  16 
units'  distance.  The  cylinder  is  Qut  by  a  plane  perpendicular  to 
V,  its  trace  passing  through  the  points  (2,0,0)  and  (14,0,16). 
Draw  the  three  projections. 


CHAPTEE  VIII. 

INTERSECTIONS  OF  CURVED  SURFACES;  CONTINUED. 

74.  Intersection  of  a  Surface  of  Revolution  and  an  Inclined 
Plane. — In  Figs.  64  and  65  a  surface  of  revolution  is  shown.    It  is 


Fig.  64. 


cut  by  an  inclined  plane  perpendicular  to  H  in  the  first  case,  and 
by  one  perpendicular  to  V  in  the  second  case.  The  planes  are 
given  by  their  traces,  and  the  problem  is  to  find  the  curves  of  inter- 
section. Both  solutions  make  use  of  cutting  planes  perpendicular 
to  PP',  the  axis  of  revolution  of  the  curved  surface. 


82 


Engineering  Descriptive  Geometry 


In  Fig.  64  a  plane  T,  taken  at  will  perpendicular  to  PP',  cuts 
the  surface  of  revolution  in  a  circular  element  seen  as  the  straight 
line  at'  in  V-  ^  is  projected  to  H  and  the  circle  aW  drawn.  The 
inclined  plane  whose  traces  are  JK  and  KL  is  intersected  by  the 
plane  T  in  a  line  whose  horizontal  projection  is  the  line  JK  itself. 
t  and  f  (on  lil)  are  therefore  the  intersections  of  the  intersections 
and  are  projected  to  the  front  elevation,  giving  points  on  the  re- 
quired line  of  intersection.  A  system  of  planes  such  as  T  defines 
points  enough  to  fully  determine  the  curve,  mit'n. 

In  Fig.  65  the  given  plane  has  the  traces  IX  and  XZ.  The  plane 
T  intersects  the  surface  of  revolution  on  the  circle  atct',  and  it 


intersects  the  plane  in  the  line  W,  seen  on  end  in  V  as  the  point  t. 
t  and  t'  in  ffi  are  points  on  the  required  curve  of  intersection,  mit'n. 

The  point  of  this  surface  of  revolution  APC  has  been  given  a 
special  name.  It  is  an  "  ogival  point."  The  generating  line  AP 
is  an  arc  of  60°,  center  at  C,  and  conversely  the  generating  line  PQ 
has  its  center  at  A.  The  shell  used  in  ordnance  is  usually  a  long 
cylinder  with  an  ogival  point.  A  double  ogival  surface  is  produced 
by  revolving  an  arc  of  120°  about  its  chord. 

75.  Intersection  of  Two  Surfaces  of  Eevolution:  Axes  Par- 
allel.— This  problem  is  illustrated  in  Fig.  QQ,  where  two  surfaces  of 


Intersections  of  Curved  Surfaces 


83 


revolution  are  shown.  A  horizontal  plane  T  cuts  both  surfaces  in 
circular  elements.  These  elements  are  drawn  in  JHI  as  circles  abed 
and  efgli.  t  and  t'  are  the  intersections  of  the  intersections.  From 
H  t  and  if  are  projected  to  V  and  S-  The  problem  in  Art.  73  is 
but  a  special  case  of  this  general  problem.  In  addition  to  the  solu- 
tion by  horizontal  planes  another  solution  is  there  possible,  due  to 
special  properties  of  the  cone  and  cylinder. 


76.  Intersection  of  Two  Surfaces  of  Revolution:  Axes  In- 
tersecting.— An  example  of  two  surfaces  of  revolution  whose  axes 
intersect  is  given  by  Fig.  67.  A  surface  is  formed  by  the  revolution 
of  the  curve  ww'  about  the  vertical  axis  PP\  and  another  surface 
by  revolving  the  curve  uQ  about  the  horizontal  axis  QQ\    The  in- 


84    -  Engineering  Descriptive  Geometry 

terscction  of  the  axes  PP'  and  QQ'  is  the  point  p.  The  peculiarity 
of  this  case  is  that  no  plane  can  cut  both  surfaces  in  circular  ele- 
ments. However,  a  sphere  described  with  the  point  of  intersection 
of  the  axes  as  a  center,  if  of  proper  size,  will  intersect  both  surfaces 
in  circular  elements.  V  is  parallel  to  both  axes  and  on  this  pro- 
jection a  circle  is  described  with  p  as  center  representing  a  sphere. 
The  radius  is  chosen  at  will.  To  keep  the  drawing  clear,  this 
sphere  has  not  been  described  on  plan  or  side  elevation,  as  it  would 
be  quite  superfluous  in  those  views. 

The  sphere  has  the  peculiarity  that  it  is  a  surface  of  revolution, 
using  any  diameter  as  an  axis.  The  curve  wiu'  and  the  semicircle 
mabn  are  in  the  same  plane  with  the  axis  PP'.  When  both  axes 
are  revolved  about  PP'y  a  and  b,  their  points  of  intersection,  gene- 
rate circular  elements,  which  are  common  to  the  sphere  and  to  the 
vertical  surface  of  revolution.  Therefore,  these  circles  are  the  in- 
tersections of  the  sphere  and  the  vertical  surface.  The  ff  and  S 
projections  of  these  circles  are  next  drawn. 

The  curve  uQ  and  the  semicircle  qcdr  are  in  the  same  plane  with 
the  axis  QQ\  When  both  axes  are  revolved  about  QQ\  their  inter- 
sections, c  and  d,  generate  circles  which  are  common  to  both  sur- 
faces, or  are  their  lines  of  intersection.  The  circle  generated  by  c 
is  drawn  in  ff  and  S,  but  that  generated  by  d  is  not  needed. 

The  three  circles  aa',  bb',  and  cc'  appear  as  straight  lines  on  V* 
but  from  them  the  points  t  and  s,  the  intersections  of  the  intersec- 
tions, are  determined.     These  are  points  on  the  required  curve  in 

V. 

The  circle  aa'  appears  as  a  circle  ata't'  in  H,  and  as  a  line  tt' 
in  S-  The  circle  cc'  appears  as  a  circle  ctc't'  in  S,  and  as  a  line  ee' 
in  H.  These  circles  intersect  in  H  at  ^  and  f,  and  in  S  at  #  and  t' 
and  s  and  5'.    These  are  points  on  the  required  curves  in  H  and  §. 

For  the  complete  solution,  a  number  of  auxiliary  spheres,  differ- 
ing slightly  in  radius,  must  be  used. 

77.  Intersection  of  a  Cone  and  a  Non-Circular  Cylinder. — A 
non-circular  cylinder  is  a  surface  created  by  a  line  which  moves 
always  parallel  to  itself,  being  guided  by  a  curve  lying  in  a  plane 
perpendicular  to  the  generating  line.  This  curve,  called  the  direc- 
trix, is  usually  a  closed  curve.  The  cross-section  of  such  a  cylinder 
is  everywhere  similar  to  the  directrix. 


UNIVERSITY 

OF 


Intersections  of  Curved  Surfaces 


85 


This  fact  may  be  utilized  to  advantage  in  some  cases.  In  Fig. 
68,  an  oblique  cone  and  a  non-circular  cylinder  intersect.  The 
directrix  of  the  cylinder  is  a  pointed  oval  curve,  abed  in  H.  Hori- 
zontal planes,  as  T'T,  intersect  the  cylinder  in  a  curve  identical  in 
shape  with  its  directrix,  so  that  its  projection  on  H  coincides  with 
the  projection  of  the  directrix  on  ff.  The  intersection  with  the 
cone  is  a  circle,  mt'tn,  and  the  intersections  of  the  intersections  are 
the  points  t. 


78.  Alteration  of  a  Curve  of  Intersection  by  a  Fillet. — In  Fig. 

69  a  hollow  cone  and  a  non-circular  cylinder,  abed  in  W,  intersect. 
On  the  left  half  the  unmodified  curve  of  intersection  is  traced  by 
the  method  of  the  preceding  article,  no  construction  lines  being 
shown  however,  as  the  case  is  very  simple.  On  the  right  half  the 
curve  is  modified  by  a  fillet  or  small  arc  of  a  circle  which  fills  in 
the  angular  groove.  The  fillet  whose  center  is  at  g  modifies  that 
point  of  the  line  of  intersection  marked  e.  The  top  of  the  circular 
arc  marks  the  point  where  an  J-J  or  g  projector  is  tangent  to  the 
surface. 
7 


86 


Engineering  Descriptive  Geometry 


The  corresponding  crest  to  the  fillet  at  other  positions  on  the 
curve  of  intersection  is  traced  as  follows :  If  a  line  drawn  through 
h  and  parallel  to  PG,  the  generating  line  of  the  cone,  is  used  as 
a  new  generator  it  will  by  its  rotation  about  PP'  create  a  new 
cone,  on  the  surface  of  which  the  required  line  of  the  crests  of  the 


Fig.  69. 


fillets  must  lie.  If  a  line  mn,  parallel  to  cc',  the  generating  line  of 
the  cylinder,  is  moved  parallel  to  cc',  and  at  a  constant  distance 
from  the  surface  of  the  non-circular  cylinder,  it  will  generate  a 
new  non-circular  cylinder  on  the  surface  of  which  the  required 
path  of  the  point  Ic  must  lie.  The  directrix  of  this  new  cylinder  is 
drawn  in  fl,  the  line  rms,  as  shown.    The  intersection  of  these  two 


Intersections  of  Curved  Surfaces 


87 


Fio.  70. 


88  Engineering  Descriptive  Geometry 

new  surfaces,  found  by  the  method  used  above  (or  by  planes  per- 
pendicular to  J-f  through  the  axis  PP'),  is  the  required  path  of  Ic 
or  the  line  which  appears  on  V  and  §.  The  line  rms,  representing 
the  same  path  on  H,  is  not  properly  a  line  of  the  drawing  and  is 
not  inked  except  as  a  construction  line. 

79.  Intersection  of  a  Helicoidal  Surface  and  a  Plane. — In  Fig. 
70  there  is  shown  a  long-pitched  screw  having  a  triple  tliread,  such 
as  is  often  employe.d  for  a  "  worm.^'  To  the  left  is  shown  a  partial 
longitudinal  section  giving  the  generating  lines.  In  V  the  con- 
cealed parts  of  the  helical  edges  are  omitted,  except  in  the  cases  of 
one  of  the  smaller  and  one  of  the  larger  edges.  The  plane  whose 
trace  on  V  is  KL  is  perpendicular  to  the  axis,  and  terminates  the 
screw  threads.  The  intersection  of  this  plane  with  the  screw 
threads  is  the  curve  of  intersection  to  be  drawn  on  H-  It  is  deter- 
mined by  passing  planes  containing  the  axis  of  the  worm.  One  of 
these  is  shown  by  its  traces  PR  and  BR'. 

From  points  a  and  h  in  the  plan  corresponding  points  are  plotted 
on  the  front  elevation,  a  falling  on  the  helix  of  small  diameter 
(extended  in  this  case),  and  &  on  the  helix  of  large  diameter.  This 
element  ab  of  the  helix  is  seen  to  pierce  the  plane  KL  at  Ic.  This 
point  Ic  is  projected  to  the  plan  and  is  one  of  the  points  on  the 
required  curve  mTcn. 

Problems  VIII. 

(For  units,  use  inches  on  blackboard  or  wire-mesh  cage,  or  small 
squares  on  cross-section  paper.) 

80.  An  anchor-ring  is  formed  by  revolving  a  circle  of  G  units 
diameter  about  a  vertical  axis,  so  that  its  center  moves  in  a  circle 
of  10  units  diameter,  center  at  Q  (8,8,8).  The  anchor-ring  is 
intersected  by  a  plane  parallel  to  V  passed  through  the  point 
A  (8,  6,  8)  and  by  another  plane  parallel  to  V  through  the  point 
B  (8,4,8).  Draw  the  projections  of  the  ring,  the  traces  of  the 
planes  and  the  lines  of  intersection. 

81.  The  same  anchor-ring  is  intersected  by  a  plane  perpendicu- 
lar to  \,  having  a  trace  passing  through  the  points  C  (0,  0,  2)  and 
B  (8,  0,  8).  Make  the  descriptive  drawing  and  show  the  true  shape 
of  the  lines  of  intersection. 


Intersections  of  Curved  Surfaces  89 

82.  The  same  anchor-ring  is  intersected  by  a  right  circular 
cylinder,  axis  P  (12,8,0),  P'  (12,8,16),  and  diameter  4  units. 
Make  the  descriptive  drawing  of  the  anchor-ring,  imagining  it  to 
be  pierced  by  the  cylinder. 

83.  An  anchor-ring  has  an  axis  P  (0,8,8),  P'  (16,8,8).  Its 
center  moves  in  a  plane  7  units  from  S,  describing  a  circle  of  8 
units  diameter.  The  radius  of  the  describing  circle  is  3  units.  It 
is  intersected  by  an  ogival  point  whose  axis  is  a  vertical  line 
Q  (7,  8,  3|),  §'  (7,  8, 16).  The  generating  line  of  the  ogival  point 
is  an  arc  of  60°,  with  center  at  (0,8,16),  and  radius  14  units. 
The  point  Q  is  the  vertex  and  the  point  Q'  is  the  center  of  the  circu- 
lar base  of  7  units  radius.  The  axes  intersect  at  ;?  (7,  8,  8).  Draw 
the  projections  and  the  line  of  intersection,  front  and  side  eleva- 
tions only. 

84.  The  line  P  (4, 13,  8),  P'  (16,  8,  8)  is  the  chord  of  an  arc  of 
90°,  whose  radius  is  9.2  units.  The  arc  is  the  generating  line  of  a 
surface  of  revolution  of  which  PP'  is  the  axis.  Draw  the  projection 
on  IHI-  Draw  the  end  view  on  an  auxiliary  plane  U  perpendicular 
to  PP',  the  trace  of  U  on  H  intersecting  OX  at  (16,  0,  0).  The 
surface  is  intersected  by  a  plane  perpendicular  to  f\  and  contain- 
ing the  line  PP'.    Draw  the  line  of  intersection  on  V- 

85.  The  same  surface  is  intersected  by  a  plane  perpendicular  to 
H  whose  trace  in  f\  passes  through  the  points  (4, 16, 0)  and 
(16,5,0).     Draw  the  line  of  intersection  on  V- 

86.  The  line  P  (3,  8,  8),  P'  (13,  8,  8)  is  the  chord  of  an  arc  of 
90°,  radius  7.07  units.  It  is  the  axis  of  revolution  of  a  surface  of 
which  the  arc  is  the  generating  line.  It  is  intersected  by  a  right 
circular  cone  having  its  vertex  at  9  (8,  8,  2),  and  center  of  base  at 
Q'  (8,  8, 12),  radius  of  base  5  units.    Draw  the  line  of  intersection. 

87.  A  non-circular  cylinder  has  its  straight  elements,  length  16 
units,  perpendicular  to  H-  The  directrix  is  a  smooth  curve 
through  the  points  A  (14,  6,  0),  B  (12,  4,  0),  C  (10,  4,  0), 
D  (8,  5,  0),  E  (5,  8,  0),  F  (2, 13,  0).  It  is  pierced  by  a  cylinder 
whose  base  is  in  \,  whose  axis  is  perpendicular  to  V  ^.t  the  point 
(8,0,8),  whose  radius  is  5  units,  and  whose  length  is  14  units. 
Find  the  line  of  intersection  in  §. 


90  Engineering  Descriptive  Geometry 

88.  The  line  P  (8,  8,  2),  P'  (8,  8, 14)  is  the  axis  of  a  right  cir- 
cular cjdinder  of  6  units  diameter.  Projecting  from  the  cylinder  is 
an  helicoidal  surface,  of  12  units  pitch,  of  which  G  (5,8,2), 
G'  (1,8,2)  is  the  generating  line.  The  helicoid  is  intersected 
by  a  plane  perpendicular  to  H  whose  trace  in  H  passes  through  the 
points  (5,0,0)  and  (16,11,0).  Draw  the  plan  and  front  eleva- 
tion of  the  cylinder  and  helicoid  and  plot  the  line  of  intersection 
with  the  plane. 

89.  The  helicoidal  surface  of  Problem  87  is  intersected  by  a  right 
circular  cylinder  whose  axis  Q  (12,  8,  2),  Q'  (12,  8, 14)  is  parallel 
to  PP\  The  radius  of  the  cylinder  is  3  units.  Draw  the  line  of 
intersection. 


CHAPTEE  IX. 
DEVELOPMENT  OF  CURVED  SURFACES. 

80.  Meaning  of  Development  as  Applied  to  Curved  Surfaces.— 

Many  curved  surfaces  may  be  developed  on  a  plane  in  a  manner 
similar  to  the  development  of  prisms  and  pyramids  explained  in 
Articles  45  and  46.  By  development,  is  meant  flattening  out, 
without  stretching  or  otherwise  distorting  the  surface.  If  a  curved 
surface  is  developed  on  a  plane  and  this  portion  of  the  plane,  called 
"  the  development  of  the  surface/'  is  cut  out,  this  development  may 


Fig.  71. 


be  bent  into  the  shape  of  the  surface  itself.  The  importance  of 
the  process  comes  from  the  fact  that  many  articles  of  sheet  metal 
are  so  made.  If  a  sheet  of  paper  is  bent  in  the  hands  to  any  fan- 
tastic shape,  it  will  always  be  found  that  through  every  point  of 
the  paper  a  straight  line  may  be  drawn  on  the  surface  in  some  one 
direction,  the  greatest  curvature  of  the  surface  at  this  point  being 
in  a  direction  at  right  angles  to  this  straight  line  element  through 
the  point.  The  surfaces  which  can  be  formed  by  twisting  a  plane 
surface  without  distortion  are  called  surfaces  of  single  curvature. 
The  curved  surfaces,  therefore,  which  are  capable  of  development 
are  only  those  which  are  surfaces  of  single  curvature  and  have 
straight  line  elements,  but  not  by  any  means  all  of  tl^se.    All  forms 


92 


Engineerixg  Descriptive  Geometry 


of  cylinders  and  cones,  right  circular,  oblique  circular,  or  non- 
circular,  may  be  developed.  The  helicoidal  surfaces,  illustrated  by 
Figs.  55  and  56,  though  having  straight  elements,  cannot  be  de- 
veloped, nor  can  the  hyperboloid  of  revolution,  a  surface  generated 
by  revolving  a  straight  line  about  a  line  not  parallel  nor  intersect- 
ing. Figs.  71  and  72  are  perspective  drawings  showing  the  process 
of  rolling  out  or  developing  a  right  circular  cylinder  and  a  right 
circular  cone. 

81.  Rectification  of  the  Arc  of  a  Circle. — In  developing  curved 
surfaces  it  frequently  happens  that  the  whole  or  part  of  the  cir- 
cumference of  a  circle  is  rolled  out  into  a  straight  line.  Since  the 
surface  must  not  be  stretched  or  compressed,  the  straight  line  must 
be  equal  in  length  to  the  arc  of  the  circle.  This  process  of  finding 
a  straight  line  equal  to  a  given  arc  is  called  rectifying  the  arc.    Xo 


This  angle  not 
to  exceed  60 


/ 


Fig.  73. 


Fig.  74. 


absolutely  exact  method  is  possible,  but  methods  are  known  which 
are  so  nearly  exact  as  to  lead  to  no  appreciable  error.  These  have 
the  same  practical  value  as  if  geometrically  perfect. 

In  Fig.  73,  AB  is  the  arc  of  a  circle,  center  at  C.  For  accurate 
work  the  arc  should  not  exceed  60°.  It  is  required  to  find  a 
straight  line  equal  to  the  given  arc.  Draw  AH,  the  tangent  at  one 
extremity,  and  draw  AB,  the  chord.  Bisect  AB  at  D.  Produce  the 
chord  and  set  oK  AE  equal  to  AD.  With  £'  as  a  center,  and  with 
EB  as  a  radius,  describe  the  arc  BF,  meeting  AH  at  F.  Then 
^F=arc  AB,  within  one-tenth  of  one  per  cent. 

In  this  figure,  and  in  the  two  following  ones,  the  arc  and  the 
straight  line  equal  to  it  are  made  extra  heavy  for  emphasis. 


Development  of  Curved  Surfaces 


93 


82.  Rectifying  a  Semicircle. — A  second  method,  applicable  par- 
ticularly to  a  semicircle,  was  recently  devised  by  Mr.  George  Pierce, 
In  Fig.  74  the  semicircle  AFB  is  to  be  rectified.  A  tangent  BC, 
equal  in  length  to  the  radius,  is  drawn  at  one  extremity.  Join  AC, 
cutting  the  circumference  at  D.  Lay  off  DE  =  DC,  and  join  BE, 
producing  BE  to  the  circumference  at  F.  Join  AF.  Then  the 
triangle  AEF,  shown  lightly  shaded,  has  its  periphery  equal  to  the 
semicircle  AFB,  within  one  twenty-thousandth  part.  The  peri- 
phery may  be  conveniently  spread  into  one  line  by  using  A  and  E 
as  centers,  and  with  AF  and  EF  as  radii,  swinging  F  to  the  left  to 
0  and  to  the  right  to  H  on  the  line  AF  extended.  GH  is  the  recti- 
fied length  of  the  semicircle. 

83.  To  Lay  Off  an  Arc  Equal  to  a  Given  Straight  Line. — This 
inverse  problem,  namely  to  lay  off  on  a  given  circle  an  arc  equal  to 


\^(jo  exceed  60" 


A  B 


B 


Fig.  75. 


Fig.  76. 


a  given  straight  line,  frequently  arises.  In  Fig.  75  a  line  AB  is 
given.  It  is  required  to  find  an  arc  of  a  given  radius  AC  equal  to 
the  given  line  AB.  At  A  erect  a  perpendicular,  making  AC  equal 
to  the  given  radius,  and  with  C  as  a  center  describe  the  arc  AF. 
On  AB,  take  the  point  B  at  one-fourth  of  the  total  distance  from 
A.  With  T>  as  center  and  DB  as  a  radius,  draw  the  arc  BF,  meet- 
ing AF  at  F.    AF  is  the  required  arc,  equal  to  AB. 

This  process  is  also  accurate  to  one-tenth  of  one  per  cent  if  the 
arc  AF  is  not  greater  than  60°.  If  in  the  application  of  this  process 
to  a  particular  case  the  arc  AF  is  found  to  be  greater  than  60°,  the 
line  AB  should  be  divided  into  halves,  thirds  or  quarters,  and  the 
operation  applied  to  the  part  instead  of  to  the  whole  line. 


94  Engineering  Descriptive  Geometry 

84.  Development  of  a  Straight  Circular  Cylinder. — In  Fig.  60 
let  the  intersecting  cylinders  represent  a  large  sheet-iron  ventilat- 
ing pipe,  with  two  smaller  pipes  entering  it  from  either  side.  Such 
a  piece  is  called  by  pipe  fitters  a  "  cross."  The  problem  is  to  find 
the  shape  of  a  flat  sheet  of  metal  which,  when  rolled  np  into  a 
cylinder,  will  form  the  surface  of  the  vertical  pipe,  with  the  open- 
ings already  cut  for  the  entrance  of  the  smaller  pipes.  Before 
developing  the  large  cylinder,  it  must  be  considered  as  cut  on  the 
straight  element  BB\  After  the  pipe  is  formed  from  the  develop- 
ment used  as  a  pattern,  the  element  BB'  will  be  the  location  of  a 
longitudinal  seam. 

A  rectangle.  Fig.  76,  is  first  drawn,  the  height  BB'  being  equal 
to  the  height  of  the  cylinder  and  the  horizontal  length  being  equal 
to  the  circumference  of  the  base  BCD  A.  (This  length  may  be  best 
found  by  Mr.  Pierce's  method,  which  gives  the  half-length,  BD.) 
On  the  drawing.  Fig.  60,  the  base  BCDA  must  be  divided  into 
equal  parts,  24  parts  being  usually  taken,  as  they  correspond  to 
■arcs  of  IS'',  which  are  easily  and  accurately  constructed  with  the 
draftsman's  triangles.  Only  6  of  these  24  parts  are  required  to  be 
actually  marked  on  Fig.  60,  as  the  figure  is  doubly  symmetrical 
and  each  quadrant  is  similar  to  the  others.  On  Fig.  76  the  line 
BCDAB  is  divided  into  24  parts  also,  the  numbering  of  the  lines 
of  division  running  from  0  to  6  and  back  to  0  for  each  half-length 
of  the  development.  In  V  of  Fig.  60,  draw  the  elements  corre- 
sponding to  the  points  of  division.  The  elemnt  IV  already  drawn 
corresponds  to  No.  4,  and  BB'  and  CC  correspond  to  Nos.  0  and  6. 
The  others  are  not  drawn  in  Fig.  60,  to  avoid  complicating  the 
figure,  but  would  have  to  be  drawn  in  practice  before  constructing 
the  development.  On  the  four  elements  which  are  numbered  4  on 
the  development.  Fig.  76,  lay  off  the  distances  Ir  equal  to  Ir  in 
Fig.  60.  On  the  two  elements.  Fig.  76,  numbered  6,  lay  off  Cc  or 
Aa  equal  to  Cc  of  Fig.  60,  and  imagine  the  proper  distances  to  be 
laid  off  on  elements  numbered  3  and  5.  Smooth  curves  through 
the  points  thus  plotted  are  the  ovals  which  must  be  cut  out  of  the 
sheet  of  metal  to  give  the  proper-shaped  openings  for  the  small 
pipes. 

When  it  is  known  in  advance  that  the  surface  of  such  a  cylinder 


Development  of  Curved  Surfaces 


95 


as  that  in  Fig.  60  must  be  developed,  it  is  often  possible  to  so 
choose  the  system  of  auxiliary  intersecting  planes  used  to  define 
the  curve  of  intersection  as  to  give  the  required  equally  spaced 
straight  elements  for  the  development. 

The  smaller  cylinder  may  be  developed  in  the  same  way.  A  new 
system  of  equally  spaced  straight  elements  would  probably  have  to 
be  chosen  for  this  cylinder. 

85.  Development  of  a  Right  Circular  Cone. — The  cone  of  Fig. 
63  has  been  selected  for  this  illustration.  Imagine  it  to  be  cut  on 
the  element  PB  and  flattened  into  a  plane.    The  surface  takes  the 


Fig.  77. 


form  of  a  sector  of  a  circle,  the  radius  of  the  sector  being  the  slant 
height  of  the  cone  (or  length  of  the  straight  element),  and  the  arc 
of  the  sector  being  equal  in  length  to  the  circumference  of  the  base 
of  the  cone.  Several  means  of  finding  the  length  of  the  arc  of  the 
sector  are  available. 

The  most  natural  method  is  to  rectify  the  circumference  of  the 
base  and  then,  with  the  slant  height  as  radius,  to  draw  an  arc  and 
to  lay  out  on  the  arc  a  length  equal  to  this  rectified  circumference. 
In  Fig.  63  suppose  that  the  semi-circumference  ABC  (in  fil)  has 
been  rectified  by  Pierce's  method.  In  Fig.  77  let  an  arc  be  drawn 
with  radius  PB  equal  to  PB  in  S?  Fig.  63,  and  from  B  draw  a 
tangent  BE  equal  to  one-half  the  rectified  length  of  the  semi-cir- 
cumference.   Find  the  arc  BC  equal  to  BE  by  the  method  of  Art. 


96  Engixeerixg  Descriptive  Geometry 

83,  Fig.  75.  BC  is  one-fourth  of  the  required  arc,  and  corresponds 
to  the  quadrant  BC  in  W,  Fig.  63.  Divide  the  arc  BC  and  the 
quadrant  BC  into  the  same  number  of  equal  parts,  numbering 
them  from  0  to  6,  if  6  parts  are  chosen.  Eepeat  the  divisions  in 
the  arc  CD  (equal  to  BC),  numbering  the  points  of  division  from 
6  down  to  0,  this  duplication  of  numbers  being  due  to  the  symmetry 
of  the  H  projection  of  Fig.  63,  about  the  line  APC.  In  Fig.  63, 
as  in  Fig.  77,  the  points  0  to  6  are  all  supposed  to  be  Joined  to  P, 
the  only  straight  elements  actually  shown  there  being  PO,  P4,  and 
P6. 

On  the  elements  P4  of  the  development  lay  off  the  true  length 
of  the  line  Pt  (and  the  true  length  of  the  line  Pf  also).  Ft  is  an 
oblique  line,  but  if  its  H  projector-plane  {Pt  in  H?  Fig.  63)  be 
revolved  up  to  the  position  Pm,  the  point  Mn  V  moves  to  m,  and 
Pm  is  the  true  length  of  Pt.  The  distance  Pg  (V?  in  Fig.  63)  is 
laid  off  on  P&  of  the  development. 

When  the  proper  distances  have  been  laid  off  on  the  elements 
P2,  P3  and  P5,  a  smooth  curve  may  be  drawn  through  the  points. 
The  sector,  with  this  opening  cut  in  it,  is  the  pattern  for  forming 
the  cone  out  of  sheet  iron  or  any  thin  material. 

If  the  ratio  of  PA  to  P'A  in  \,  Fig.  63,  can  be  exactly  deter- 
mined, the  most  accurate  method  of  getting  the  angle  of  the  sector  is 
by  calculation,  for  the  degrees  of  arc  in  the  development  are  to  the 
degrees  in  the  base  of  the  cone  (360°)  as  the  radius  of  the  base  of 
the  cone  is  to  the  slant  height.  In  this  case  P'A  is  f  PA.  The 
sector  in  Fig.  75  subtends  f  X360°,  or  216°.  In  the  use  of  this 
method  a  good  protractor  is  required  to  lay  out  the  arc. 

Problems  IX. 

90.  Draw  an  arc  of  60°  with  10  units  radius.  At  one  end  draw 
a  tangent  and  on  the  tangent  lay  off  a  length  equal  to  the  given 
arc.  On  the  tangent  lay  off  a  length  of  8  units,  and  find  the  length 
of  arc  equal  to  this  distance. 

91.  An  arc  of  12  units  radius,  one  of  9  units  radius,  and  a 
straight  line  are  all  tangent  at  the  same  point.  Find  on  the  tan- 
gent the  straight  line  equal  in  length  to  45°  of  the  large  arc.  Find 
the  length  on  the  other  arc  equal  to  this  length  on  the  tangent  and 
show  that  it  is  an  arc  of  60°. 


Development  of  Curved  Surfaces        97 

92.  Rectify  a  semicircle  of  10  units  radius  and  compare  this 
length  with  the  calculated  length,  31.4  units. 

93.  A  rectangle  31.4  units  by  12  units  is  the  developed  area  of  a 
cylinder  of  10  units  diameter.  A  diagonal  line  is  drawn  on  the 
development,  which  is  then  rolled  into  cylindrical  form.  Plot  the 
form  taken  by  the  diagonal  and  show  that  it  is  a  helix  of  12  units 
pitch. 

94.  A  right  circular  cone  has  a  base  of  10  units  diameter,  and 
a  vertical  height  of  12  units.  Its  slant  height  is  13  units.  Calcu- 
late the  angle  of  the  sector  which  is  the  developed  surface  of  the 
cone.  Find  this  angle  by  rectifying  the  circumference  of  the  base 
of  the  cone,  and  by  finding  the  arc  equal  to  the  rectified  length. 
(This  last  operation  must  be  performed  on  one-third  or  one-quarter 
of  the  rectified  length,  to  keep  the  accuracy  within  one-tenth  of 
one  per  cent.) 

95.  A  semicircle,  radius  10  units,  is  rolled  up  into  a  cone.  What 
is  the  radius  of  the  base?  What  is  the  slant  height?  What  is  the 
relation  between  the  area  of  the  curved  surface  of  the  cone  and  the 
area  of  the  base  ? 

96.  A  right  circular  cylinder,  such  as  Fig.  49,  is  of  7.59  units 
diameter,  and  12  units  height.  It  is  intersected  by  a  plane  per- 
pendicular to  V  through  the  points  C  and  A'.  Draw  plan,  front 
elevation  and  the  development  of  the  surface. 

97.  A  right  circular  cone,  like  that  of  Fig.  51,  has  its  front  ele- 
vation an  equilateral  triangle,  each  side  being  10  units  in  length. 
From  Av  a  perpendicular  is  drawn  to  PvCv  cutting  it  at  E.  If  this 
line  represents  a  plane  perpendicular  to  V?  draw  the  development 
of  the  cone  with  the  line  of  intersection  of  the  cone  and  plane  traced 
on  the  development. 

98.  A  right  circular  cylinder,  standing  in  a  vertical  position,  as 
in  Fig.  49,  diameter  7  units,  and  length  10  units,  is  pierced  from 
side  to  side  by  a  square  hole  3 J  units  on  each  edge,  the  axis  of  the 
hole  and  the  axis  of  the  cylinder  bisecting  each  other  at  right 
angles.    Draw  the  development  of  the  surface. 

99.  A  sheet  of  metal  22  units  square  with  a  hole  11  units  square 
cut  out  of  its  middle,  the  sides  of  the  hole  being  parallel  to  the 
edges  of  the  sheet,  is  rolled  up  into  a  cylinder.  Draw  the  plan, 
front  and  side  elevations  of  the  cylinder. 


CHAPTER  X. 

STRAIGHT  LINES  OF  UNLIMITED  LENGTH  AND  THEIR 

TRACES. 

86.  Negative  Coordinates. — ^We  have  dealt  only  with  points  hav- 
ing positive  or  zero  coordinates,  and  the  lines  and  planes  have  been 


limited  in  their  extent,  or,  if  infinite,  have  extended  indefinitely 
only  in  the  positive  directions.  As  it  becomes  necessary  at  times 
to  trace  lines  and  planes  in  their  course,  no  matter  if  they  cross 
the  reference  planes  into  new  regions  of  space,  the  use  and  meaning 
of  negative  coordinates  must  be  explained.  The  value  of  the  x 
coordinate  of  a  point  is  the  length  of  the  §  projector  or  perpen- 
dicular distance  from  the  point  to  the  side  reference  plane  g.  (See 
Figs.  6  and  7,  Art.  9.)     If  this  value  decreases  gradually  to  zero, 


Lines  of  Unlimited  Length:    Their  Traces 


99 


the  point  moves  towards  S  until  it  lies  in  S  itself.  If  this  value 
becomes  negative,  it  is  clear  that  the  point  crosses  the  side  reference 
plane  into  a  space  to  the  right  of  it. 

For  example,  a  point  P,  having  a  variable  x  coordinate,  but  hav- 
ing its  y  coordinate  always  equal  to  4  and  its  z  coordinate  equal  to 
2,  is  a  point  moving  on  a  line  parallel  to  the  axis  of  X.  If  x  de- 
creases to  zero,  it  is  on  S  at  the  point  marked  Ps  in  Fig.  78.  If 
the  X  coordinate  decreases  further,  reaching  a  value  of  —3,  it 
moves  to  the  point  P  in  that  figure.  Fig.  78  is  the  perspective 
(  —  3,4,2).  The  y  and  z  projectors  cannot 
H  and  V  in  their  customary  positions,  but 


drawing  of  a  point  P 
project  the  point  P  to 


•  IH],ext.t\ 

j  \ 

X      '  0 

V  9 

z 

and      ^v 
^extended. 

\ 


E 

e,   Js 


:ti 


9  p 


H 


Y.    ^ 


i5       V 


\ 


h 


s 


Fig.  79. 


Fig.  80. 


Fig.  81. 


project  it  upon  parts  of  those  planes  extended  beyond  the  axes  of 
Y  and  Z,  as  shown.  In  Fig.  79,  the  corresponding  descriptive 
drawing,  it  must  be  understood  that  the  plane  IM,  extended,  has 
been  revolved  with  H?  about  the  axis  of  X,  into  the  plane  of  the 
paper,  V?  and  §  has  been  revolved  as  usual  about  the  axis  of  Z, 
coming  into  coincidence  with  V?  extended.  This  "  development " 
of  the  planes  of  reference  is  exactly  as  described  in  Art.  7.  It  is 
noticeable  that  the  x  coordinate  of  P  is  laid  off  to  the  right  of  the 
origin  instead  of  to  the  left.  Ph  lies,  therefore,  in  the  quadrant 
which  usually  represents  no  plane  of  projection,  and  Pv  lies  in  the 
quadrant  which  usually  represents  S-  Ps  lies  in  its  customary 
place,  since  both  y  and  z,  the  coordinates  which  alone  appear  in  S> 
are  positive. 


100  Engineering  Descriptive  Geometry 

It  is  evident  that  the  laws  of  projection  for  ff,  V  ^^d  §,  Art. 
11,  have  not  been  altered,  but  simply  extended.  Ph  and  Pv  are  in 
the  same  vertical  line;  Pv  and  Ps  are  in  the  same  horizontal  line; 
and  the  construction  which  connects  Pn  and  Ps  still  holds  good. 

In  Fig.  79  the  space  marked  S  represents  not  only  S  but  V 
extended  as  well. 

In  Fig.  80  is  represented  a  point  P  (3,  —  2,  3 ) ,  having  a  negative 
y  coordinate.  The  point  is  in  front  of  V^  ^t  2  units'  distance,  not 
behind  V-  The  projection  on  fi,  instead  of  being  above  the  axis 
of  X  a  distancee  of  2  units,  is  below  it  by  the  same  amount.  So  also 
the  projection  on  S  is  to  the  left  of  the  axis  of  Z,  a  distance  of  2 
units,  instead  of  the  the  right  of  it.  After  developing  the  reference 
planes  in  the  manner  of  Art.  7,  plane  H?  extended,  has  come  into 
coincidence  with  V?  a^d  plane  §,  extended,  has  also  come  into  co- 
incidence with  V-  Thus  the  field  representing  V  represents  also 
the  other  two  reference  planes,  extended. 

In  Fig.  81  a  point  P  (2,2,-3)  having  a  negative  z  coordinate 
is  represented.  The  point  is  above  H  3  units,  instead  of  below  ff, 
at  the  same  perpendicular  distance.  P  projects  upon  V  on  V 
extended  above  the  axis  of  X.  After  developing  the  reference 
planes,  plane  ff  comes  into  coincidence  witti  V  extended.  Pa  is 
on  §  extended  above  the  axis  of  Y,  and  therefore  after  develop- 
ment it  occupies  the  so-called  "  construction  space." 

Points  having  two  or^three  negative  coordinates  may  be  dealt 
with  in  the  same  manner,  but  are  little  likely  to  arise  in  practice. 

It  is  evident  that  subscripts  must  be  used  invariably,  to  prevent 
confusion  whenever  negative  values  are  encountered. 

87.  Graphical  Connection  Between  Ph  and  Ps- — In  Figs.  79,  80 
and  81,  Ph  and  Ps  are  connected  by  a  construction  line  PhfhfsPs  in 
a  manner  which  is  an  extension  of  that  shown  by  Fig.  7,  Art.  9. 
Xote  that  the  quadrant  of  a  circle  connecting  Ph  and  Ps  must  be 
described  alw^ays  on  the  construction  space  or  on  the  field  devoted 
to  V,  never  on  the  fields  devoted  to  IM  or  S- 

88.  Traces  of  a  Line  of  Unlimited  Length,  Parallel  to  an  Axis. — 
A  straight  line  w^hich  has  no  limit  to  its  length,  but  extends  in- 
definitely in  either  direction,  must  necessarily  have  some  points 
whose  coordinates  are  negative.    In  passing  from  positive  to  nega- 


Lines  of  Unlimited  Length:    Their  Traces 


101 


tive  regions  the  line  must  pass  through  some  plane  of  reference 
(having  one  of  its  coordinates  zero  at  that  point),  and  the  point 
where  it  pierces  a  plane  of  reference  is  called  the  trace  of  the  lino 
on  that  plane  of  reference,  the  word  trace  being  used  to  indicate  a 
"  track ''  or  print  showing  the  passage  of  the  line. 

Lines  parallel  to  the  axes  have  been  used  freely  already.  An  fil 
projector  is  simply  a  vertical  line  or  line  parallel  to  the  axis  of  Z. 
Any  perspective  figure  showing  a  point  P  and  its  horizontal  pro- 
jection Ph  will  serve  as  an  illustration  of  this  line,  as  PPh  in  Fig. 
G,  Art.  9. 


Fig.  82. 


Imagine  PPh  to  be  extended  in  both  directions  as  an  unlimited 
straight  line.  Then  Pn  is  the  trace  of  the  line  on  H.  In  Fig.  7, 
the  point  Ph  itself  is  the  W  projection  of  the  line.  Pt-<?.  extended 
in  both  directions,  is  the  vertical  projection  and  Psfs  is  the  side 
projection.  Thus  it  is  seen  that  a  vertical  line  has  but  one  trace, 
that  on  the  plane  to  which  it  is  perpendicular.  PPv  may  be  taken 
as  an  illustration  of  a  line  parallel  to  the  axis  of  Y,  and  PPg  of  one 
parallel  to  the  axis  of  X.  A  better  example  of  this  latter  case  is 
shown  iH^igs.  15  and  16,  Art.  16.  The  line  BAAs,  perpendicular 
to  S,  has  itfe  trace  on  §  at  Aa, 


% 


r 


102 


Engineering  Descriptive  Geometry 


89.  Traces  of  an  Inclined  Straight  Line. — An  inclined  line  snch 
as  AB  in  Figs.  82  and  83  pierces  two  reference  planes  as  at  A  and 
B,  but  as  it  is  parallel  to  the  third  reference  plane,  S,  it  has  no 
trace  on  S.  The  peculiarity  of  the  descriptive  drawing  of  this  line. 
Fig.  83,  is  the  apparent  coincidence  of  the  H  and  V  projections 
as  one  vertical  line.  The  S  projection  is  required  to  determine  the 
traces  A  and  B. 

90.  Traces  of  an  Oblique  Straight  Line :  The  H  and  V  Traces. — 
An  oblique  line,  if  unlimited  in  length,  must  pierce  each  of  the 
reference  planes,  since  it  is  oblique  to  all  three.     Any  line  is  com- 


FiG.  84. 


Fig.  85. 


Fig.  86. 


Fig.  87. 


pletely  defined  when  two  points  on  the  line  are  given.  If  two 
traces  of  a  straight  line'  are  given,  the  third  trace  cannot  be  assumed, 
but  must  be  constructed  from  the  given  conditions  by  geometrical 
process.  It  will  always  be  found  that  of  the  three  traces  of  an 
oblique  line  one  trace  at  least  has  some  negative  coordinate. 

As  the  complete  relation  between  the  three  traces  is  somewhat 
complicated,  the  relation  between  two  traces,  as,  for  instance,  H 
and  V  traces,  must  be  considered  first.  Two  cases  are  shown,  the 
first  by  Figs.  84  and  85,  and  the  second  by  Figs.  86  and  87.  The 
line  AB  is  the  line  whose  traces  are  A  (5,0,4)  and  B  (2,4,0). 
The  line  CD  is  the  line  whose  traces  are  C  (7,  0,  5)  and  D  (2, 4, 0). 


Lines  of  Unlimited  Length:    Their  Traces 


103 


From  the  descriptive  drawing  of  AB,  Fig.  85,  it  is  seen  that  the 
H  projection  of  the  line  cuts  the  axis  of  X  vertically  above  the 
trace  on  V^  ^^^  that  the  V  projection  cuts  the  axis  of  X  vertically 
under  the  trace  on  H.  It  may  be  noted  that  the  two  right  triangles 
AhBBv  and  BvAAn  have  the  line  AhB^  on  the  axis  of  X  as  their 
common  base.  From  the  descriptive  drawing  of  the  line  CD,  Fig. 
87,  it  is  seen  that  the  effect  of  the  vertical  trace  C  having  a  nega- 
tive z  coordinate  simply  puts  C  (on  V)  above  Cn,  instead  of  below  it. 
The  two  right  triangles  CnDDv  and  DvCCh  have  the  line  ChDv  on 
the  axis  of  X,  as  their  common  base,  but  the  latter  triangle  is  above 
the  axis  instead  of  in  its  normal  position. 


Fig.  88. 


91.  Traces  of  an  Oblique  Straight  Line:  The  V  and  S  Traces. — 

Figs.  88  and  89  show  two  lines  piercing  V  aii4  S- 

The  line  AB  pierces  V  at  A  and  S  at  B.  The  two  right  triangles 
AsABv  and  BvBAg  have  their  common  base  AgBv  on  the  axis  of  Z. 
The  line  CD  pierces  V  at  (7  and  S  extended  at  D,  the  point  D 
having  a  negative  y  coordinate.  The  right  triangles  CgCDv  and 
DvDCs  have  their  base  DvCg  in  common  on  the  axis  of  Z,  but  in  the 
descriptive  drawing  DvDCs  lies  to  the  left  of  the  axis  of  Z  instead 
of  to  the  right,  owing  to  the  point  D  having  a  negative  y  coordinate. 


104 


Engineerixg  Descriptive  Geometry 


92.  Traces  of  an  Oblique  Straight  Line :  The  H  and  S  Traces.— 

Figs.  90  and  91  show  two  lines  piercing  ff  and  §. 

The  line  AB  pierces  n  at  A  and  §  at  B.  The  triangles  AsABh 
and  BhBAs  have  their  common  base  AsBn  on  the  axis  of  Y,  Fig.  90, 
but  in  the  descriptive  drawing  the  duplication  of  the  axis  of  Y 
causes  this  base  AsBh  to  separate  into  two  separate  bases,  one  on 
OYh  and  one  on  OYs.    Otherwise,  there  has  been  no  change. 

The  line  CD  pierces  H  at  C  and  S  extended  at  D,  the  point  D 
having  a  negative  z  coordinate.    In  Fig.  90  CsCDh  and  DhDCs  have 


Fig.  90 


their  common  base  CgDh  on  the  axis  of  Y,  but  in  the  descriptive 
drawing  CgDh  appears  in  two  places.  The  triangle  DnDCs  lies  above 
S  in  the  "construction  space,"  or  on  S  extended,  since  D  has  a 
negative  z  coordinate. 

93.  Three  Traces  of  an  Oblique  Straight  Line. — Figs.  92  and  93 
show  an  oblique  straight  line  ABC  piercing  V  at  A,  H  at  B,  and 
S  extended  at  C.  Since  the  line  is  straight,  the  three  projections 
of  the  line  AB^Cv,  AsBsC  and  AjtBCh  are  all  straight  lines.  In  the 
perspective  drawing.  Fig.  92,  part  of  the  V  projection  is  on  V  ex- 
tended and  part  of  the  S  projection  on  S  extended. 


Lines  of  Unlimited  Length:    Their  Traces 


105 


In  the  descriptive  drawing,  Fig.  93,  the  relation  between  A  and 
B  is  the  same  as  that  in  Fig.  85,  as  shown  by  the  two  triangles 
AhABv  and  BvBAh,  or  the  quadrilateral  AhABvB.  The  relation 
between  A  and  C,  as  shown  by  the  quadrilateral  AgACvC,  is  the 
same  as  that  between  A  and  B,  Fig.  89,  as  shown  by  the  quadri- 
lateral AsABvB.  The  relation  between  B  and  C,  Fig.  93,  as  shown 
by  the  two  triangles  BsBCn  and  ChCBg,  is  the  same  as  that  between 
C  and  D  of  Fig.  91,  as  shown  by  the  triangles  CgCDh  and  DhDCs. 
Xo  new  feature  has  been  introduced. 


Small  part 
o/Y  extended 


Fig.  92. 


94.  Paper  Box  Diagram. — To  assist  in  understanding  Figs.  92 
and  93,  a  model  in  space  should  be  made  and  studied  from  all 
sides.  The  complete  relation  of  the  traces  is  then  quickly  grasped. 
Construct  the  descriptive  drawing,  Fig.  93,  on  coordinate  paper, 
using,  as  coordinates  for  A,  B  and  C,  (15,0,12),  (5,12,0),  and 
(0, 18,  —6).  Fold  into  a  paper  box  after  the  manner  of  Fig.  9, 
Art.  12,  having  first  cut  the  paper  on  some  such  line  as  mn,  so  that 
the  part  of  the  paper  on  which  C  is  plotted  may  remain  upright, 
serving  as  an  extension  to  g.  It  will  be  found  that  a  straight  wire 
or  long  needle  or  a  thread  may  be  run  through  the  points  A,  B  and 
C,  thus  producing  a  model  of  the  line  and  all  its  projections. 


106  Engineering  Descriptive  Geometry 

95.  Intersecting  Lines. — If  two  lines  intersect,  their  point  of 
intersection,  when  projected  upon  any  plane  of  reference,  must 
necessarily  be  the  point  of  intersection  of  the  projections  on  that 
plane.  For  example,  a  line  AB  intersects  a  line  CD  at  E.  Project 
E  upon  a  plane  of  reference,  as  lil.  Then  Eh  must  be  the  point  of 
intersection  of  AhBn  and  ChDh.  In  the  same  way  Ev  must  be  the 
point  of  intersection  of  AvBv  and  CvDv,  and  Eg  of  AgBg  and  CsDg. 

To  determine  whether  two  lines  given  by  their  projections  meet 
in  space  or  pass  without  meeting,  the  projections  on  at  least  two 
reference  planes  must  be  extended  (if  necessary)  till  they  meet. 
Then  for  the  lines  themselves  to  intersect,  the  points  of  intersec- 
tion of  the  two  pairs  of  projections  must  obey  the  rules  of  pro- 
jection of  a  point  in  space  (Art.  11).  Thus  if  AhBh  and  ChDn  are 
given  and  meet  at  a  point  vertically  above  the  point  of  intersection 
of  AvBv  and  CvDv,  the  two  lines  really  meet  at  a  point  whose  pro- 
jections are  the  intersections  of  the  given  projections.  If  this  con- 
dition is  not  filled  the  lines  pass  without  meeting,  the  intersecting 
of  the  projections  being  deceptive. 

96.  Parallel  Lines. — If  two  lines  are  parallel,  the  projections  of 
the  lines  on  a  reference  plane  are  also  parallel  (or  coincident). 
For,  the  two  lines  make  the  same  angle  with  the  plane  of  pro- 
jection; their  projector-planes  are  parallel;  and  the  projections 
themselves  are  parallel. 

Thus  if  a  line  AB  is  parallel  to  another  line  CD,  then  AhBh  must 
be  parallel  to  ChDh,  AvBv  to  CvDv,  and  AgBg  to  CsDg.  If  the  two 
lines  lie  in  a  plane  perpendicular  to  a  plane  of  projection — for 
example,  perpendicular  to  H — then  the  ff  projector-planes  coin- 
cide and  the  ff  projections  also  coincide.  The  V  and  S  projections 
are  parallel  but  not  coincident. 

If  two  lines  do  not  fill  the  conditions  of  intersecting  or  of  parallel 
lines,  they  must  necessarily  be  lines  which  pass  at  an  angle  without 
meeting. 


Lines  of  Unlimited  Length:    Their  Traces         107 

Problems  X. 

100.  Plot  the  points  4  (8,  6, -4),  5  (7, -3,  5),  (7  (-7,  0, 12). 

101.  Plot  the  points  A  (6,  -10,  3),  5  (0,0,-8),  C  (-6,5,4). 

102.  Make  a  descriptive  drawing  of  a  line  26  units  long  from 
the  point  P  (—8,4,9),  perpendicular  to  g.  What  traces  does  it 
have  ?    What  are  the  coordinates  of  its  middle  point  ? 

103.  A  line  is  drawn  from  P  (12,5,16)  perpendicular  to  H. 
Make  the  descriptive  drawing  of  the  line,  and  of  a  line  perpen- 
dicular to  it,  drawn  from  Q  (0,  0,  8).  What  is  the  length  of  this 
perpendicular  line,  and  where  are  its  traces? 

104.  A  straight  line  extends  from  A  (8, 12, 0)  through  D 
(8,  6,  8)  for  a  distance  of  20  units.  Make  the  descriptive  drawing 
of  the  line.    Where  are  its  traces  and  its  middle  point  ? 

105.  A  straight  line  pierces  H  at  A  (8,  6, 0)  and  Y  at  B 
(8,  0, 12).  Draw  its  projections.  Where  is  its  trace  on  §?  What 
are  the  coordinates  of  D,  its  middle  point  ? 

106.  A  straight  line  extends  from  E  (15,6,16)  through  A 
(3,  6,  0)  to  meet  g.  Make  the  descriptive  drawing  and  mark  the 
traces  on  H  and  §. 

107.  Draw  the  lines  A  (16,  11,  8),  B  (4,  8,  2)  ;  C  (12,  5,  10), 
D  (0,2,4);  and  E  (11,3,0),  F  (5,15,8).  Which  pair  meet, 
which  are  parallel,  and  which  pass  at  an  angle?  What  are  the 
coordinates  of  the  point  of  intersection  of  the  pair  which  meet? 

108.  The  points  A  (8,  0, 12),  B  (0,  8,  6)  and  (7  (-8, 16,  0)  are 
the  traces  of  a  straight  line.  Make  the  descriptive  drawing  of  the 
line. 

109.  The  points  A  (8,  -4,0),  D  (4,4,6)  and  E  (2,8,9)  are 
on  a  straight  line.  Find  the  trace  B  where  it  pierces  V  and  the 
trace  C  where  it  pierces  S. 


CHAPTER  XI. 
PLANES  OF  UNLIMITED  EXTENT:  THEIR  TRACES. 

97.  Traces  of  Horizontal  and  Vertical  Planes. — The  lines  of 
intersection  of  a  plane  with  the  reference  planes  are  called  its 
traces.  Planes  of  unlimited  extent  may  be  of  three  kinds,  parallel 
to  a  reference  plane,  inclined,  or  oblique.  Unlimited  planes  of  the 
first  two  classes  have  been  dealt  with  already,  but  for  the  sake  of 
precision  may  be  treated  here  again  to  advantage. 

A  horizontal  plane  is  one  parallel  to  fi,  and  the  trace  of  such  a 
plane  on  V  is  a  line  parallel  to  the  axis  of  X,  and  the  trace  on  § 
is  a  line  parallel  to  the  axis  of  Y.  These  traces  meet  the  axis  of  Z 
at  the  same  point  and  appear  on  the  descriptive  drawing  as  one 
continuous  line.  There  is.  of  course  no  trace  on  H-  In  Pig-  58, 
Art.  67,  the  plane  T,  represented  by  its  traces  TT  on  V  and  TT" 
on  §,  is  a  horizontal  plane.  These  traces  are  not  only  the  intersec- 
tions of  T  with  H  and  S,  but  T  is  "  seen  on  edge  "  in  those  views.- 
Every  point  of  the  plane  T ,  when  projected  upon  V^  lies  somewhere 
on  the  line  T'T,  extended  indefinitely  in  either  direction. 

A  vertical  plane  parallel  to  V  ^las  for  its  traces  a  line  on  ff 
parallel  to  the  axis  of  X,  and  on  S  a  line  parallel  to  the  axis  of 
Z,  with  no  trace  on  V-  These  traces  meet  the  axis  of  Y  at  the 
same  point,  and  appear  on  the  descriptive  drawing  as  two  lines  at 
right  angles  to  this  axis,  the  point  on  Y  separating  into  two  points 
as  usual.  In  Fig.  57,  Art.  ^Qi,  a  vertical  plane  U,  parallel  to  V?  is 
represented  by  its  traces  WR  on  H  and  RU"  on  g. 

A  vertical  plane  parallel  to  §  has  for  its  trace  on  H  a  line  paral- 
lel to  the  axis  of  Y ,  and  for  its  trace  on  V  a  line  parallel  to  the 
axis  of  Z,  with  no  trace  on  g.  These  traces  meet  the  axis  of  X  at 
the  same  point  and  appear  on  the  descriptive  drawing  as  one  con- 
tinuous line. 

98.  Traces  of  Inclined  Planes. — Inclined  planes  are  those  per- 
pendicular to  one  reference  plane,  but  not  to  two  reference  planes. 
The  auxiliary  planes  of  projection  have  been  of  this  kind.     In 


Planes  of  Unlimited  Extent:    Their  Traces 


109 


Fig.  20,  Art.  22,  the  plane  \],  perpendicular  to  Hi,  has  the  line 
MX  for  its  trace  on  ff,  and  XN  for  its  trace  on  V-  I^-  the  de- 
scriptive drawing.  Fig.  21,  MX  and  XNv  are  these  traces. 

If  in  Fig.  20  both  U  and  S  are  imagined  to  be  extended  towards 
the  eye,  they  will  intersect  in  a  line  parallel  to  OZ.  This  §  trace 
will  be  on  S  extended,  and  every  point  of  it  will  have  the  same 
negative  y  coordinate.  Of  the  three  traces  of  U>  two  are  vertical 
lines,  and  one  only,  MX,  is  an  inclined  line.  The  plane  in  Fig.  64, 
Art.  7-1,  may  be  taken  as  a  second  example  of  an  inclined  plane 
perpendicular  to  ff.  The  trace  on  §  is  not  a  negative  line  in  this 
case,  but  is  a  vertical  line  on  §  to  the  right  of  the  axis  of  Z  at  a 
distance  equal  to  OJ. 

In  Fig.  57,  Art.  66,  IJ,  JK  and  KL  are  the  three  traces  of  an 
inclined  plane  perpendicular  to  V-  ^^  every  case  of  an  inclined 
plane  the  inclined  trace  is  on  that  reference  plane  to  which  it  is 
perpendicular,  and  shows  the  angles  of  the  inclined  plane  with  one 
or  both  of  the  other  reference  planes. 

p\  ^    b] 

X 


Fig.  95. 


99.  Traces  of  an  Oblique  Plane:  All  Traces  "Positive." — The 

general  case  of  an  oblique  plane  is  shown  in  Fig.  94.  The  plane 
P  is  represented  as  cutting  the  cube  of  reference  planes  in  the  lines 
marked  PH,  PV  and  PS.  These  lines  are  the  traces  of  the  plane 
P,  and  may  be  understood  to  extend  indefinitely,  the  plane  itself 
extending  in  all  directions  without  limit.  They  are  shown  limited 
in  Fig.  94  in  order  to  make  a  more  realistic  appearance.  PH,  PV, 
and  PS  are  used  to  define  the  three  traces. 


110 


Engineering  Descriptive  Geometry 


Where  PH  and  PV  meet  we  have  a  point  common  to  three  planes, 
P,  HI  and  V-  Since  it  is  common  to  fil  and  V  it  is  on  the  line  of 
intersection  of  H  and  V>  or  in  other  words  it  is  on  the  axis  of  X. 
This  point  is  marked  a.  In  the  same  way  PH  and  P8  meet  at  h 
on  the  axis  of  Y,  and  PV  and  PS  meet  at  c  on  the  axis  of  Z. 

The  descriptive  drawing,  Fig.  95,  is  obvious  from  the  explana- 
tion of  the  perspective  drawing.  From  Fig.  95  it  is  evident  that 
if  two  traces  of  a  plane  are  given  the  third  trace  can  be  determined 


Fig.  96. 


Fig.  97. 


by  geometrical  construction.  Thus,  if  PH  and  PV  are  given,  P8 
may  be  defined  by  extending  PH  to  &  on  the  axis  of  Y  and  extend- 
ing PV  to  c  on  the  axis  of  Z.  The  line  joining  he  is  the  required 
trace  of  the  plane  on  §.  If  any  two  points  on  one  trace  are  given, 
and  any  one  point  on  a  second  trace,  the  whole  figure  may  be  com- 
pleted. Thus  any  two  points  on  PH  define  that  line  and  enable  a 
and  h  to  be  found.  A  third  point  on  PV,  taken  in  conjunction 
with  a,  defines  PV,  and  enables  c  to  be  located.  Ic,  as  before, 
defines  the  trace  PS.  This  is  an  application  of  the  general  prin- 
ciple that  three  points  determine  a  plane. 


Planes  of  Unlimited  Extent:    Their  Traces 


111 


100.  Traces  of  an  Oblique  Plane:  One  Trace  "Negative." — In 

Figs.  94  and  95  the  plane  P  lias  been  so  selected  that  all  traces  have 
positive  positions.  These  are  the  portions  usually  drawn.  Of 
course  each  trace  may  be  extended  in  either  direction,  points  on 
the  trace  then  having  one  or  more  negative  coordinates.  Any 
trace  having  points  all  of  whose  coordinates  are  positive,  or  zero, 
may  be  called  a  positive  trace. 

In  Fig.  96  a  plane  P  is  shown,  intersecting  H  and  V  in  the 
"positive"  traces  PH  and  PV.  The  third  trace,  P8,  in  this  case, 
has  no  point  all  of  whose  coordinates  are  positive.  In  the  descrip- 
tive drawing,  Fig.  97,  the  tw^o  positive  traces,  meeting  at  a  on  the 


\ 

b 

f 

^yfo.  Y3 

s 

Fig.  98. 


Fig.  99. 


axis  of  Xj  are  usually  considered  as  fully  representing  the  plane  P. 
From  these  lines  PH  and  PV,  alone,  the  imagination  is  relied  upon 
to  "  see  the  plane  P  in  space,"  as  shown  by  Fig.  96. 

In  Fig.  98,  the  plane  Q  is  represented.  Ordinarily  the  positive 
traces  QV  and  ^^S",  meeting  at  c  on  the  axis  of  Z,  are  the  only 
traces  shown  in  the  descriptive  drawdng,  Fig.  99,  and  are  considered 
to  indicate  perfectly  the  path  of  the  plane  Q. 

101.  Position  of  the  Negative  Trace. — The  negative  trace  PS, 
in  Fig.  96,  is  shown  as  one  of  the  edges  of  the  rectangular  plate 
representing  the  unlimited  plane  P.  This  line  PS  has  been  de- 
termined by  extending  PH  to  meet  the  axis  of  Y  (extended)  at 


112  EXGINEERIXG    DESCRIPTIVE    GEOMETRY 

h,  and  by  extending  PV  to  meet  the  axis  of  Z  (extended)  at  c. 
The  line  joining  h  and  c  is  the  trace  PS.  It  will  be  noted  that  in 
finding  the  location  of  Pas'  in  Fig.  97,  PV  has  been  extended  to  cut 
the  axis  of  Z  (extended  np  from  ZO)  at  c  and  PH  has  been  ex- 
tended to  cut  the  axis  of  Y  (extended  down  from  YO)  at  h.  1) 
has  been  rotated  90°  about  the  origin,  and  the  points  h  and  c  thus 
plotted  (on  S  extended)  have  been  found  to  give  the  line  PS. 
Every  step  of  the  process  and  the  lettering  of  the  figure  have  been 
similar  to  those  used  in  finding  PS  from  PH  and  PF  in  Art.  98. 

In  Fig.  98,  the  negative  trace  is  QH,  the  top  line  of  the  rect- 
angular plate  representing  the  unlimited  plane  Q.  QH  has  been 
determined  as  follows:  QV  extended  meets  the  axis  of  X  extended 
at  a,  and  QS  extended  meets  the  axis  of  Y  extended  at  h.  The  line 
ab  is  therefore  the  trace  on  H,  or  QH.  In  the  descriptive  drawing 
the  same  process  of  extending  QV  to  a  and  QS  to  h  determines  the 
line  QH,  a  line  every  point  of  which  has  some  negative  coordinate. 
Of  course  QH  must  be  considered  as  drawn  on  parts  of  the  plane 
lil  extended  over  V^  S?  and  the  so-called  construction  space.  In 
finding  the  negative  traces,  it  is  imperative  to  letter  the  diagrams 
uniformly,  keeping  a  for  the  intersection  of  the  plane  with  the  axis 
of  X,  h  for  that  with  the  axis  of  Y,  and  c  for  that  with  the  axis  of 
Z.  With  this  rule  h  will  always  be  the  point  which  is  doubled  by 
the  separation  of  the  axis  of  Y  into  two  lines,  and  the  arc  hh  will 
always  be  described  in  the  construction  space  or  in  the  quadrant 
devoted  to  V?  never  in  those  devoted  to  H  and  g. 

102.  Parallel  Planes. — If  two  planes  are  parallel  to  each  other, 
their  traces  on  H,  V  and  S  are  parallel  each  to  each.  This  prop- 
osition may  be  proved  as  follows:  If  we  consider  two  planes  P 
and  Q  parallel  to  each  other  and  each  intersecting  the  plane  H,  the 
lines  of  intersection  with  ff  (PH  and  QH)  cannot  meet,  for,  if 
they  did  meet,  the  planes  themselves  would  meet  and  could  not  then 
be  parallel  planes.  PH  and  QH  must  therefore  be  parallel  lines 
described  on  H.  Thus,  if  a  plane  P  and  a  plane  Q  are  parallel, 
then  PH  and  QH  are  parallel,  PV  and  QV  are  parallel,  and  PS 
and  QS  are  parallel. 

The  method  of  finding  the  true  length  of  a  line  by  its  projection 
upon  a  plane  parallel  to  itself,  treated  in  Chapter  III,  is  really  the 


Planes  of  Unlimited  Extent:    Their  Traces 


113 


process  of  passing  a  plane  parallel  to  a  projector-plane  of  the  given 
line.  Thus  in  Fig.  21,  Art.  22,  the  auxiliary  plane  U  has  its  hori- 
zontal trace  XM  parallel  to  AhBn,  and  the  vertical  trace  of  the  ff 
projector-plane,  if  drawn,  would  be  parallel  to  XNv. 

103.  The  Plane  Containing  a  Given  Line. — If  a  line  lies  on  a 
plane,  the  trace  of  the  line  on  any  plane  of  reference  (the  point 
where  it  pierces  the  plane  of  reference)  must  lie  on  the  trace  of  the 
plane  on  that  plane  of  reference.  Thus,  if  the  line  EF,  Fig.  100, 
lies  on  the  plane  P,  then  A,  the  trace  of  EF  on  ff,  lies  on  PH,  the 
trace  of  P  on  H ;  and  B,  the  trace  of  EF  on  V,  lies  on  PV,  the 
trace  of  P  on  V. 


Fig.  100. 


Fig.  101. 


From  this  fact  it  follows  that  to  pass  a  plane  which  will  contain 
a  given  line  it  is  necessary  to  find  two  traces  of  the  line  and  to  pass 
a  trace  of  the  plane  through  each  trace  of  the  line.  As  an  infinite 
number  of  planes  may  be  passed  through  a  given  line,  it  is  neces- 
sary to  have  some  second  condition  to  define  a  single  plane.  For 
example,  the  plane  may  be  made  also  to  pass  through  a  given  point 
or  to  be  perpendicular  to  a  reference  plane. 

In  Fig.  100,  if  only  the  line  EF  is  given  and  it  is  required  to  pass 
a  plane  P,  containing  that  line,  and  containing  also  some  point, 
as  a,  on  the  axis  of  X,  the  process  is  as  follows:  Extend  the  line 
EF  to  A  and  B,  its  traces  on  HI  and  V-    Joi^i  Ba  and  oA.    These 


114  Engineering  Descriptive  Geometry 

are  the  traces  of  the  required  plane  P.  In  the  descriptive  drawing, 
Fig,  101,  the  corresponding  operation  is  performed.  A  and  B 
must  be  determined  as  in  Art.  90,  and  joined  to  a.  These  lines 
represent  the  traces  of  a  plane  containing  the  line  EF  and  the 
chosen  point  a. 

To  pass  a  plane  Q  containing  the  line  EF  and  also  perpendicular 
to  H  (Figs.  100  and  101),  the  trace  of  §  on  H  must  coincide  with 
the  projection  of  EF  on  H,  for  the  required  plane  perpendicular  to 
fi  is  the  ff  projector-plane  of  the  line.  Its  traces  are  therefore 
ABh  and  BnB, 

The  traces  of  a  plane  containing  EF  and  perpendicular  to  V  are 
BAv  and  AvA. 

104.  The  Line  or  Point  on  a  Given  Plane. — ^To  determine  whether 
a  line  lies  on  a  given  plane  is  a  problem  the  reverse  of  that  just 
treated.  It  amounts  sim.ply  to  determining  whether  the-  traces  of 
the  line  lie  on  the  traces  on  the  plane.  Thus,  in  Fig.  101,  if  PV 
and  PH  are  given,  and  the  line  EF  is  given  by  its  projections,  the 
traces  of  EF  must  be  found,  and  if  they  lie  on  PH  and  PV  the  line 
is  then  known  to  lie  on  the  given  plane  P. 

To  determine  whether  a  given  point  lies  on  a  given  plane  is 
almost  as  simple.  Join  one  projection  of  the  point  with  any  point 
on  the  corresponding  trace  of  the  plane.  Find  the  other  trace  of 
the  line  so  formed,  and  see  whether  it  lies  on  the  other  trace  of  the 
given  plane.  Thus  in  Fig.  101,  if  the  traces  PH  and  PV  and  the 
projections  of  any  one  point,  as  E,  are  given,  select  some  point  on 
PH,  as  A,  and  join  EjA  and  EvAv.  Find  the  trace  B.  If  it  lies 
on  PV,  the  point  E  itself  lies  on  P. 

To  draw  on  a  given  plane  a  line  subject  to  some  other  condition, 
such  as  parallel  to  some  plane  of  reference,  is  always  a  problem  in 
constructing  a  line  whose  traces  are  on  the  traces  of  the  given 
plane,  and  which  yet  obeys  the  second  condition,  whatever  it  may  be. 

105.  The  Plane  Containing  Two  Given  Lines. — From  the  last 
article,  if  a  plane  contains  tivo  given  lines,  the  traces  of  the  plane 
must  contain  the  traces  of  the  lines  themselves.  The  given  lines 
must  be  intersecting  or  parallel  lines,  or  the  solution  is  impossible. 

In  Fig.  102  two  lines,  AB  and  AC,  are  given  by  their  projections. 
They  intersect  at  A,  since  Ah,  the  intersection  of  the  fl  projections, 


Planes  of  Unlimited  Extent:    Their  Traces 


115 


is  vertically  above  Av,  the  intersection  of  the  V  projections.  Ex- 
tend the  lines  to  E,  F,  G  and  H,  their  traces  on  ff  and  V-  Join 
the  fi  traces,  E  and  G,  and  produce  the  line  also  to  a  on  the  axis 
of  Z.  Join  the  V  traces,  H  and  F,  and  extend  the  line  HF  also 
to  a.  Ea  and  aH  are  the  traces  of  a  plane  P  containing  both  lines, 
AB  and  AC.  The  meeting  of  the  two  traces  at  a  is  a  test  of  the 
accuracy  of  the  drawing. 

This  process  may  be  applied  to  a  pair  of  parallel  lines,  but  not  of 
course  to  two  lines  which  pass  at  an  angle  without  meeting. 


-L     1 

Y 

\>xG 

^ 

^0 

-^       V 

z 

Fig.  102. 


106.  The  Line  of  Intersection  of  Two  Planes. — If  two  planes 
P  and  Q  are  given  by  their  traces,  their  line  of  intersection  must 
pass  through  the  point  where  the  Qi  traces  meet  and  the  point 
where  the  V  traces  meet.  Thus,  in  Fig.  103,  FB.  and  QB.  meet 
at  A  and  FY  and  QY  meet  at  B.  A  and  B  are  points  on  the 
required  line  of  intersection  of  F  and  Q,  and  since  A  is  on  H  and 
B  is  on  Vj  they  are  the  H  and  V  traces  of  the  line  of  intersection. 
AB}i  and  BAv  are  therefore  the  projections,  and  should  be  marked 
FQn  and  FQ^, 


.116 


Engineering  Descriptive  Geometry 


A.™    " 

V         >^   . 

Fig.  103. 


Planes  of  Unlimited  Extent:    Their  Traces        117 

107.  Special  Case  of  the  Intersection  of  Two  Planes :  Two  Traces 
Parallel. — ^The  construction  must  be  varied  a  little  in  the  special 
case  when  two  of  the  traces  of  the  planes  are  parallel.  In  Fig.  104 
the  traces  PV  and  QV  are  parallel.  In  carrj'ing  out  the  construc- 
tion as  in  Fig.  100,  it  is  necessary  to  join  Av  with  B.  But  the 
point  B  is  the  intersection  of  PV  and  QV,  which  are  parallel,  and 
is  therefore  a  point  at  an  infinite  distance  in  the  direction  of  those 
lines,  as  indicated  by  the  bracket  on  Fig.  104.  To  join  Av  with  B 
at  infinity  means  to  draw  a  line  through  Av  parallel  to  PV  and  QV. 


PH  ^ 

\ 

QH 

^ 

r\  \ 

^\ 

•\N   \ 

^                POv 

0 

Ja^  % 

'^% 

PV 

A    S 

Qvy 

z 

Fig.  105. 


From  B,  at  infinity,  a  perpendicular  must  be  supposed  to  be  drawn 
to  the  axis  of  X,  intersecting  it  at  Bh.  Bn  is  therefore  at  an  infinite 
distance  to  the  right  on  the  axis  of  X  (extended).  To  join  the 
point  A  with  the  point  Bh  means,  therefore,  to  draw  a  line  through 
4  parallel  to  the  axis  of  X.  These  lines  are  the  required  projec- 
tions of  PQ. 

108.  Special   Case   of  the  Intersection   of   Two   Planes:   Four 

Traces  Parallel. — Another  special  case  arises  when  the  four  traces 

(on  two  planes  of  projection)  are  parallel.    It  is  then  necessary  to 

refer  to  a  third  plane  of  projection.    In  Fig.  105  the  planes  P  and 

9 


118 


Engineering  Descbiptive  Geometry 


Q  have  their  four  traces  on  H  and  V  all  parallel.  The  planes  are 
inclined  planes  perpendicular  to  §,  and  if  their  traces  are  drawn 
on  S,  their  intersection  is  the  line  PQ.  In  S  both  P  and  Q  are 
"  seen  on  edge/'  so  their  line  of  intersection  is  "  seen  on  end." 
From  PQsf  PQv  and  PQn  are  drawn  by  projection. 

109.  The  Point  of  Intersection  of  a  Line  and  a  Plane. — The 
simple  cases  of  this  problem  have  been  previously  explained  and 
used.     If  the  plane  is  horizontal,  vertical  or  inclined,  there  is 


Fig.  106. 


always  one  view  at  least  in  which  it  is  seen  on  edge.  In  that  view 
the  given  line  is  seen  to  pierce  the  given  plane  at  a  definite  point 
from  which,  by  the  rules  of  projection,  the  other  views  of  the  point 
of  intersection  are  easily  determined.  Thus  in  Fig.  27,  Art.  38, 
the  point  a,  where  PA  pierces  the  plane  KL,  is  determined  first  in 
V  and  then  projected  to  H  and  §. 

The  general  case  of  this  problem  may  be  solved  as  in  Fig.  106. 
A  plane  P  is  given  by  its  traces  PH  and  PV.  A  line  AB  is  given 
by  its  projections.    It  is  required  to  find  where  AB  pierces  P.    The 


Planes  of  Unlimited  Extent:    Their  Traces        119 

solution  is  as  follows:  Let  a  plane  perpendicular  to  V  be  passed 
through  the  projection  AvBv.  x\ccording  to  Art.  103  the  traces  of 
this  plane  are  BvFv  and  FvF.  Draw  the  line  of  intersection  of  this 
plane  with  the  plane  P  (Art.  106)  as  follows:  BvFv  and  PV  in- 
tersect at  E.  F  and  E  are  the  traces  of  the  line  of  intersection  of 
the  two  planes.  Complete  the  drawing  of  the  line  of  intersection 
in  H,  as  FEh.      ■ 

Referring  to-  the  horizontal  projection,  AnBn  is  seen  to  intersect 
FEh,  the  lil  projection  of  the  line  of  intersection,  at  Wh.  Since 
both  FE  and  AB  are  lines  which  lie  in  the  vertical  projector-plane 
through  AB,  this  point  of  intersection,  Wn,  is  the  projection  of  the 
true  point  of  intersection,  W,  of  those  two  lines.  From  Wh  project 
to  Wv  for  the  other  projection  of  W.  This  point  W  which  lies  on 
P  and  is  on  the  line  AB  is  the  required  point. 

Problems  XI. 

(For  blackboard  or  cross-section  paper  or  wire-mesh  cage.) 

110.  Plot  the  point  A  (4,7,9).  Pass  a  horizontal  plane  P 
through  the  point  A,  and  draw  the  traces  of  P.  Also  a  vertical 
plane  Q,  parallel  to  V^  and  draw  its  traces.  Also  an  inclined  plane 
R,  perpendicular  to  H,  making  an  angle  of  45°  with  OX. 

111.  Plot  the  line  A  (8,2,4),  B  (2,6,16).  Pass  an  inclined 
plane  P  perpendicular  to  H  through  this  line  and  draw  the  traces 
of  P.  At  C,  the  middle  point  of  AB,  pass  a  plane  Q  perpendicular 
to  P  and  to  H,  and  draw  QH  and  QV. 

112.  The  plane  P  cuts  the  axes  at  the  points  a  (10,0,0), 
I  (0,  5,  0)  and  c  (0,  0, 15).  Pass  a  plane  Q  parallel  to  P,  through 
the  point  a'  (6,  0,  0). 

113.  A  plane  P  has  its  trace  on  H  through  the  points 
A  (12,12,0)  and  h  (0,6,0).  Its  trace  on  V  passes  through  the 
point  c  (0,  0, 12).  Draw  the  three  traces.  Draw  three  traces  of  a 
plane  Q,  parallel  to  P  through  the  point  c'  (3,  0,  0). 

114.  An  indefinite  line  contains  the  points  A  (11,2,6)  and 
B  (5,6,0).  Pass  a  plane  P  perpendicular  to  H  containing  this 
line  and  draw  the  traces  PH,  PV  and  PS.  Pass  a  plane  Q  con- 
taining this  line  and  the  point  a'  (2,0,0).  Draw  the  traces  QII 
and  QV.    Draw  the  negative  trace  QS  on  §  extended  over  H. 


120  Engineering  Descriptive  Geometry 

115.  A  plane  P  cuts  the  axis  of  Z  at  a  (4,  0,  0),  the  axis  of  Y 
at  &  (0,  6,  0),  and  the  axis  of  Z  at  c  (0,  0,-12).  Draw  its  traces. 
Draw  the  V  and  S  traces  of  a  plane  Q  parallel  to  P  and  containing 
the  line /i  (1,4, 11),  5  (4,1,14). 

116.  An  inclined  plane,  perpendicular  to  W,  has  for  its  V  and 
§  traces  lines  parallel  to  OZ  at  positive  distances  of  15  and  5  units. 
An  inclined  plane  Q  perpendicular  to  H  has  its  V  and  §  traces 
parallel  to  OZ  at  distances  of  12  units  and  8  units.  Draw  all  three 
traces  and  the  projections  of  PQ,  their  line  of  intersection. 

117.  Draw  the  traces  of  a  plane  P,  containing  the  points 
A  (8, 1,  3),  B  (4,  5, 1)  and  C  (2,  4,  3).  Does  the  point  D  (4, 1,  5) 
lie  on  this  plane? 

118.  The  traces  of  a  plane  P  are  lines  through  the  points 
a  (10,0,0),  h  (0,15,0)  and  E  (14,0,6).  A  plane  Q  has  its 
traces  through  the  points  a'  (2,0,0),  E,  and  F  (7,5,0).  Draw 
the  projections  of  their  line  of  intersection,  PQ. 

119.  The  plane  P  cuts  the  axes  at  a  (12,  0,  0),  h  (0, 12,  0)  and 
c  (0,  0, 12).  Where  does  the  line  K  (1,  5, 12),  L  (5,  3,  6)  pierce 
the  plane? 


CHAPTER  XII. 


VARIOUS  APPLICATIONS*. 

110.  Traces  of  an  Inclined  Plane  Perpendicular  to  an  Oblique 
Plane. — One  of  the  most  general  devices  used  in  the  drafting  room 
is  the  auxiliary  plane  of  projection,  and  it  is  often  advantageous 
to  pass  this  plane  perpendicular  to  some  plane  of  the  drawing  in. 


UlS 


^ 


\ 


>  f 

^\ 

.  '^ 

V 

z 

c 

Fig.  107. 


Fig.  108.  Fig.  109. 


order  to  get  the  advantage  of  showing  that  plane  ''  on  edge."  Thus 
in  Fig.  31,  Art.  42,  the  plane  U  has  been  taken  perpendicular  to 
the  long  rectangular  faces  of  the  triangular  prism,  in  order  to 
show  clearly  where  BB'  and  DD'  pierce  those  planes.  The  manner 
of  passing  the  plane  U  was  fairly  clear  in  that  case  from  the 
simplicity  of  the  figure.  However,  as  it  is  not  always  clear  how  to 
pass  a  plane  perpendicular  ,to  an  oblique  plane,  the  general  method 
may  well  be  explained  here.  In  Fig.  107  the  plane  P,  previously 
shown  in  Fig.  94,  is  represented,  and  an  auxiliary  plane  l!J>  Per- 
pendicular to  it  and  to  H?  is  shown.  The  traces  of  P  are  PH,  PV 
and  PS  as  before,  and  the  traces  of  U  are  UH  and  US,    It  must 


122  Engineering  Descriptive  Geometry 

be  understood  that  the  ff  traces  of  these  planes,  PH  and  UH,  are 
perpendicular  to  each  other,  as  this  condition  is  essential  if  P  and 
HJ  are  to  be  planes  perpendicular  to  each  other. 

Fig.  108  is  the  descriptive  drawing  corresponding  to  the  per- 
spective drawing.  Fig.  107.  At  some  point  h  on  PH  a  line  Mdh 
has  been  drawn  perpendicular  to  PH.  This  line  is  the  inclined 
trace  of  a  plane  U  perpendicular  to  H.  The  other  traces  of  U  are 
parallel  to  the  axis  of  Z  (Art.  98).  One  of  these,  the  trace  on§, 
is  shown  by  the  line  dgNg,  parallel  to  OZ,  dn  and  dg  being  two 
representations  of  the  same  point  d  in  Fig.  107,  just  as  In  and  &, 
represent  the  point  h,  duplicated.  Mdn  may  be  called  UH  and  dsN^ 
may  be  called  IJ8.  UH  and  US  are  the  traces  of  an  inclined  plane 
\],  perpendicular  to  the  oblique  plane  P. 

The  proof  that  P  and-  U  are  perpendicular  to  each  other  is  as 
follows:  If,  in  Fig.  107,  a  line  hh'  is  drawn  perpendicular  to  H 
at  the  point  h,  it  will  lie  in  the  plane  JJ-  The  angle  ahh'  will 
then  be  an  angle  of  90°,  and  by  construction  the  angle  ahd  is  also 
90°.  Thus  the  line  ah  is  perpendicular  to  two  intersecting  lines  de- 
scribed in  the  plane  \]  and  is  therefore  perpendicular  to  U  itself. 
The  plane  P  contains  the  line  PH  and  is  thus  perpendicular  to  \}, 

111.  An  Auxiliary  Plane  of  Projection  Perpendicular  to  an 
Oblique  Plane. — To  utilize  the  inclined  plane  U  as  an  auxiliary 
plane  of  projection,  its  developed  position  must  be  shown  by  drawing 
dhNu  perpendicular  to  UH,  This  line  is  the  duplicate  position  of 
dsNs  or  US.  In  developing  the  planes,  U  is  first  revolved  on  UH 
as  an  axis  into  the  plane  of  H  as  shown  in  Fig.  109,  and  then  with 
H  into  the  plane  of  the  paper,  V-  The  trace  of  P  on  \},  or  PU,  is 
the  line  of  intersection  of  the  planes,  and  is  shown  clearly  in  Fig. 
107.  This  line  passes  through  h  where  PH  and  UH  meet,  and 
through  5  where  PS  and  US  meet.  In  Fig.  108,  dhS  is  laid  off  on 
dfj^u,  equal  to  dgS,  and  the  line  hs  is  the  required. trace  of  P  on  U> 
or  PU.  The  actual  line  PU ,  in  Fig.  108,  is  only  that  part  of  PU, 
in  Fig.  107,  which  is  between  h  and  s,  shown  as  a  broken  line. 

The  important  part  in  this  process  is  that  \]  is  taken  perpen- 
dicular to  P,  so  that  P  is  "  seen  on  edge  "  on  U-  By  this  process 
the  plane  P,  which  is  ohlique  when  ff,  V  and  S  are  considered. 


Various  Applications 


123 


becomes  an  inclined  plane  when  only  ff  and  \]  are  considered. 
As  it  is  easier  to  deal  with  inclined  than  with  oblique  planes,  we 


u 


/i^ 


Fig.  110. 


may  now  treat  P  as  inclined  to  ff  and  perpendicular  to   \J  in 
further  operations. 

Fig.  108  is  well  adapted  to  making  a  paper  box  diagram  which, 


124  Engineerixg  Descriptive  Geometry 

when  folded,  will  give  most  of  the  lines  of  Fig.  107.  To  reconstruct 
Fig.  108,  plot  the  points  a  (18,0,0),  I  (0,18,0),  c  (0,12,0), 
d  (0,6,0),  h  (6,12,0)  and  s  (0,6,8).  The  line  dr,Nu  is  at  an 
angle  of  45°  with  ZOYn  and  the  construction  space  YsOdjiNu  can 
be  folded  away  inside  by  creasing  or  cutting  it  on  several  lines. 

112.  Intersection  of  an  Oblique  Plane  and  a  Cylinder. — An  ex- 
ample of  the  use  of  an  auxiliary  view  on  which  an  oblique  plane  iti 
seen  on  edge  is  shown  in  Fig.  110.  An  inclined  cylinder  is  inter- 
sected by  an  oblique  plane  P  given  by  its  traces  PH,  PV  and  PS. 
It  is  required  to  describe  on  the  cylinder  the  curve  of  intersection 
of  the  plane  and  the  cylinder.  The  solution  is  as  follows:  An 
auxiliary  plane  \J,  perpendicular  to  P  and  to  Qi,  is  chosen,  and 
PU  is  drawn  upon  \]  as  in  Fig.  108.  PU  is  the  view  of  P  "  seen  on 
edge  "  in  U-  Auxiliary  cutting  planes  parallel  to  Ji  are  used  for 
the  determination  of  the  required  line  of  intersection.  The  traces 
of  one  of  the  planes  are  drawn,  as  TT  in  V,  Tr  in  S,  and  T'T'" 
in  U-  This  latter  trace  is  parallel  to  duM  (or  UH),  because  T  is 
parallel  to  H,  and  the  distance  dhT"  is  equal  to  d^T'  in  S-  T"r" 
cuts  the  axis  of  the  cylinder  at  p.  p  is  projected  to  H,  and  the 
circular  element  described  in  H,  with  p  as  a  center,  is  the  inter- 
section of  the  auxiliary  plane  T  and  the  cylinder.  In  HJ  the  planes 
P  and  T  are  both  "  seen  on  edge,"  intersecting  in  a  line  "  seen  on 
end."  This  point  projected  to  ff  gives  this  line  of  intersection  of 
P  and  T  as  tf. 

The  intersections  of  the  intersections  are  therefore  the  points  t 
and  f ,  where  the  circle  and  the  straight  line  meet. 

113.  The  Angle  between  Two  Oblique  Lines. — This  problem  of 
finding  the  angle  between  two  oblique  lines  is  shown  in  Fig.  111. 
Let  two  lines  AB  and  AC,  meeting  at  A,  be  given  by  their  If  and 
V  projections.    It  is  required  to  find  the  true  angle  between  them. 

By  the  process  of  Art.  105,  Fig.  102,  the  traces  of  the  plane  con- 
taining AB  and  AC  are  found  and  the  lines  are  all  lettered  accord- 
ing to  Fig.  102. 

An  auxiliary  plane  of  projection,  U,  is  passed  perpendicular  to 
PV,  and  therefore  perpendicular  to  both  P  and  V?  and  is  revolved 
into  the  plane  V-     The  projections  of  AB  and  AC  on  this  plane 


Various  Applications 


125 


fall  in  the  single  line  AuCuBu,  since  P,  the  plane  of  the  lines,  is 
"  seen  on  edge  "  on  \].  A  portion  of  the  plane  P  is  now  revolved 
about  the  JJ  projector  of  the  point  A  into  a  position  parallel  to 
XM.  In  Uj  Cu  moves  to  C'u  and  j5„  to  B'u,  revolving  about  A  as 
their  center.  In  Y>  Bv  moves  to  B'v  and  Cv  to  C'v,  both  parallel  to 
XM.  This  is  the  process  of  finding  the  true  length  of  a  line  by 
revolving  about  a  projector,  as  in  Art.  32.    AvB'v  is  the  true  length 


Fig.  111. 


of  AB;  AvC'v  is  the  true  length  of  AC ;  and  B'vAvC'v  is  the  true 
angle  between  the  lines. 

This  process  makes  it  possible  to  find  the  true  shape  of  any 
figure  described  on  an  oblique  plane. 

114.  A  Plane  Perpendicular  to  an  Inclined  Line. — It  is  often 
advantageous  to  pass  a  plane  perpendicular  to  a  line  in  order  to 
use  the  plane  as  a  plane  of  projection,  on  which  the  given  line  will 
be  seen  on  end  as  a  point.  The  method  of  passing  a  plane  perpen- 
dicular to  an  inclined  line  is  shown  in  Fig.  112.  Let  AB  be  an 
inclined  line,  lying  in  a  plane  parallel  to  V;  so  that  AnBh  is  parallel 


126 


Engineering  Descriptive  Geometry 


to  the  axis  of  X,  It  is  required  to  find  the  traces  of  a  plane  P 
perpendicular  to  AB.  The  essential  point  is  that  the  traces  of  the 
plane  must  be  perpendicular  to  the  corresponding  projections  of 
the  line.  Thus,  choose  some  point  p  on  the  inclined  projection  of 
the  line,  in  this  case  on  AvB^,  and  through  p  draw  a  perpendicular 
to  AvBv,  to  serve  as  the  trace  of  F.     At  a,  where  this  trace  FY 


B, 


S 
^ 


Fig.  112. 


meets  the  axis  of  X,  erect  a  perpendicular  to  FTL.  These  lines  FY 
and  FK  are  the  traces  of  an  inclined  plane  perpendicular  to  AB 
and  to  V-  It  is  noticeable  that  the  inclined  trace  of  the  plane  is 
on  that  reference  plane  which  shows  the  inclined  projection  of  the 
line.* 


*  A  proof  that  F  is  perpendicular  to  AB  is  as  follows :  AB 
is  the  line  of  intersection  of  its  own  H  projector-plane,  and  its 
own  V  projector-plane.    F  is  perpendicular  to  both  these  projector- 


Various  Applications 


127 


115.  Application  of  a  Plane  Perpendicular  to  a  Line. — In  Fig. 
113  an  application  of  an  inclined  plane  perpendicular  to  an  in- 
clined line  is  made  for  -the  purpose  of  finding  the  line  of  intersection 
between  an  inclined  cone  and  an  inclined  cylinder  whose  axes  do 
not  meet. 


Fig.  113. 


If  from  P,  the  vertex  of  the  cone,  a  line  Pp  is  drawn  parallel  to 
QQ',  as  shown,  any  plane  which  contains  this  line  and  cuts  both 

planes.  For,  P  is  perpendicular  to  V  and  therefore  to  the  W  pro- 
jector-plane, which  is  parallel  to  V;  the  V  projector-plane  is  per- 
pendicular to  V.^  so  that  it  is  seen  on  edge  on  V  just  as  is  P  itself ; 
apAv  is  therefore  the  true  angle  between  these  two  planes,  and  by 
construction  is  a  right  angle.  P  is  therefore  perpendicular  to  both 
projector-planes  and  therefore  to  the  line  AB,  which  is  their  line 
of  intersection. 


128  Engineering  Descriptive  Geometry 

surfaces  will  cut  only  simple  elements  of  the  surfaces.  Tor  such 
a  plane  contains  the  vertex  of  the  cone,  and  therefore,  if  it  cuts 
the  cone,  will  cut  it  in  straight  elements;  and  such  a  plane  is 
parallel  to  QQ',  and  therefore,  if  it  cuts  the  cylinder,  cuts  only 
straight  elements.  No  other  planes  can  be  found  which  cut  simple 
elements  and  can  be  used  to  determine  the  line  of  intersection. 

If  a  plane  U  is  passed  perpendicular  to  Pp  at  any  point  p,  and 
is  used  as  an  auxiliary  plane  of  projection,  Pp  will  be  seen  on  end 
as  the  point  P,  and  any  plane  R  through  P,  seen  on  edge  in  \},  as 
RR\  will  cut  only  straight  elements  on  the  two  curved  surfaces. 
The  complete  projections  of  the  cone  and  cylinder  have  been  shown 
on  \]y  and  the  plane  R  cuts  the  bases  at  a,  h,  c  and  d.  These  points 
projected  to  V  enable  the  elements  to  be  drawn  there,  and  the 
intersections  of  the  intersections  are  the  four  points  marked  r. 
From  V  these  points  are  projected  to  W  and  g.  Two  of  these 
points  r  have  been  projected  to  the  other  views  to  show  the  neces- 
sary construction  lines. 

116.  A  Plane  Perpendicular  to  an  Oblique  Line. — To  pass  a 
plane  perpendicular  to  an  oblique  line,  it  is  only  necessary  to  draw 
the  traces  of  the  plane  perpendicular  to  the  corresponding  pro- 
jections of  the  line.  In  Fig.  114,  let  AB  be  an  oblique  line.  At 
any  point  on  AhBh  draw  a  perpendicular  line  PH.  From  a,  where 
PH  meets  the  axis  of  X,  draw  PV  perpendicular  to  AB* 

A  paper  box  diagram  traced  from  Fig.  114,  or  constructed  on 
cWdinate  paper,  using  the  coordinates  A  (10,  4,  4)  and  B  (6,  8,  2), 
C  (2, 12,  0)  and  D  (14,  0,  6),  and  a  (8,  0,  0),  will  assist  materially 
in  understanding  the  problem. 

The  oblique  plane  P  is  not  serviceable  as  an  auxiliary  plane  of 
projection. 

117.  The  Application  of  Axes  of  Projection  to  Mechanical 
Drawings. — Descriptive  Geometry  is  a  geometrical  science,  the 
science  dealing  primarily  with  orthographic  projectioi*-,  while  Me- 
chanical Drawing  is  the  art  of  applying  these  principles  to  the 

*  The  proof  of  this  construction  is  more  difficult  than  in  the 
corresponding  case  of  an  inclined  line,  but  it  depends  as  before 
on  the  line  AB  being  the  intersection  of  its  H  and  V  projector- 
planes,  and  these  planes  themselves  being  perpendicular  to  P, 


Various  Applications 


129 


needs  of  engineers  and  mechanics  in  the  pursuit  of  industries. 
Mechanical  Drawing  includes  therefore  many  abbreviations  and 
conventional  representations,  which  seek  to  curtail  unnecessary 
work  and  often  to  convey  information  as  to  methods  of  manu- 
facture and  other  such  commercial  considerations  foreign  to  the 
strict  scientific  study. 


Fig.  114. 


In  Mechanical  Drawing  many  lines  necessary  to  the  strict  execu- 
tion of  a  descriptive  drawing  are  omitted  as  unnecessary  to  the 
application  of  the  principles,  when  once  the  principles  have  been 
fully  grasped.  A  noteworthy  omission  is  the  axes  of  projection, 
which,  though  absent,  still  govern  the  rules  for  making  the  draw- 
ing.   Instead  of  measuring  distances  from  the  axes  for  every  point 


130  Engineering  Descriptive  Geometry 

on  the' drawing,  the  "center  lines"  of  the  different  views  (which 
really  represent  central  planes)  are  laid  off  and  distances  from 
these  center  lines  are  thereafter  used.  This  is  the  regular  pro- 
cedure in  drawing-room  practice.  That  this  difference  is  purely- 
one  of  omission  is  clear  from  the  fact  that  axes  of  projection  may 
always  be  inserted  in  a  mechanical  drawing.  If  two  views  only  of 
a  piece  are  presented,  any  line  between  them  (perpendicular  to  the 
lines  of  projection  from  one  view  to  another)  may  be  selected  as 
the  axis  of  X,  and  any  convenient  point  on  that  line  as  the  origin 
of  coordinates. 

If  three  views  are  given,  as,  for  example,  Fig.  32,  Art.  44,  sup- 
posing the  axes  to  be  there  omitted,  a  ground  line  XOYg  may  be 
selected  at  will,  dividing  the  fields  of  H  and  V-  The  other  line 
must  be  determined  as  follows :  By  the  dividers  take  the  vertical 
distance  from  OX  to  the  center  line  mn,  and  lay  off  this  distance 
horizontally  to  the  left  from  the  center  line  of  the  side  elevation. 
The  line  ZOYn  may  then  be  drawn.  All  y  coordinates  of  points  will 
now  check  correctly,  measured  parallel  to  the  two  axes  of  Y,  if  the 
original  drawing  itself  is  accurate. 

It  is  thus  evident  that  in  applying  Descriptive  Geometry  to  prac- 
tical mechanical  drawing  we  may  fall  back  upon  the  use  of  axes  of 
projection  whenever  the  lack  of  them  is  felt. 

118.  Practical  Application  of  Descriptive  Geometry. — Many 
draftsmen  have  picked  up  a  knowledge  of  Descriptive  Geometry 
without  direct  study  of  the  science.  This  is  largely  due  to  the  fact 
that,  till  very  recently,  all  books  on  Descriptive  Geometry  were 
based  on  a  system  of  planes  of  projection  which  are  analogous  to 
the  methods  of  practical  drawing  in  use  on  the  continent  of  Europe, 
but  which  are  little  used  in  England,  and  hardly  at  all  in  the  United 
States  of  America.  It  will  be  found,  however,  that  in  American 
drafting  rooms  all  the  usual  devices  of  draftsmen  are  applications, 
sometimes  almost  unconscious  applications,  of  the  principles  covered 
in  the  preceding  chapters.  The  favorite  device  is  the  application 
of  an  inclined  auxiliary  plane  of  projection,  suitably  chosen;  next 
in  importance  is  the  rotation  of  the  object  to  show  some  true  shape ; 
while  other  applications  are  used  less  frequently.  The  methods  of 
determining  lines  of  intersection  of  planes  and  curved  surfaces  are 
exactly  those  described  in  Chapters  IV,  VII  and  VIII. 


Various  Applications  131 

Problems  XII. 

(For  use  on  blackboard,  with  cross-section  paper  or  wire-mesh. 

cage.) 

120.  The  plane  P  has  its  traces  through  the  points  a  (14,  0,  0), 
&  (0,14,0)  and  c  (0,0,7).  Pass  a  plane  Q,  perpendicular  to  F 
and  to  H,  through  the  point  J.  (5,  7,  0).  If  §  is  to  be  used  as  an 
auxiliary  plane  of  projection,  draw  the  trace  of  P  on  §  when  Q  has 
been  revolved  into  coincidence  with  H. 

121.  Draw  the  traces  of  a  plane  P  cutting  the  axes  at  the  points 
a  (12,0,0),  I  (0,8,0)  and  c  (0,0,12).  Draw  the  traces  of  an 
auxiliary  plane,  U?  perpendicular  to  FlI  at  the  point  A  (3,  6,  0). 
Is  the  point  B  (6, 1,  4^)  on  the  plane  P? 

122.  The  fi  trace  of  a  plane  P  passes  through  the  points 
^  (12,  5,  0)  and  P  (6,  2,  0).  Its  V  trace  passes  through  (7  (9,  0,  6). 
Pass  an  inclined  plane  perpendicular  to  H  and  perpendicular  to 
T,  through  the  point  P  (5,  9,  7). 

123.  Of  a  plane  F,  PH,  the  horizontal  trace,  passes  through  the 
points  A  (5,3,0)  and  B  (13,9,0),  and  FV  passes  through 
C  (12,  0, 11).  Complete  the  traces  of  P  and  draw  the  traces  of  a 
plane  perpendicular  to  FV  at  the  point  B  (9,  0,  8).  Prove  that  the 
line  E  (9,  6, 1),  F  (7,  3,  2)  lies  on  the  plane  P. 

124.  A  sphere  of  radius  7  units  has  its  center  at  C  (8,  8,  8).  A 
plane  P  cuts  the  axes  of  projection  at  a  (26,  0,  0),  h  (0, 13,  0)  and 
c  (0,0,13).  Pass  an  auxiliary  plane  of  projection  \},  perpen- 
dicular to  H  and  to  F,  cutting  the  axis  oi  X  oi  d  (16,  0,  0).  Draw 
the  trace  of  P  on  \].  The  circle  of  intersection  of  the  sphere  and 
the  plane  P  is  seen  on  edge  in  U-  Show  the  elliptical  projection 
of  this  circle,  on  H,  by  passing  auxiliary  cutting  planes  parallel 
to  U-  (If  this  problem  is  solved  by  use  of  wire-mesh  cage,  the 
point  a  is  inaccessible,  but  FH  passes  through  E  (16,5,0),  and 
FV  through  F  (16,0,5).     The  plane  S'  can  be  turned  to  serve 

asU) 

125.  Find  the  true  shape  of  the  triangle  A  (3,  2,  6),  B  (9,  6,  2), 
C  (8,  0, 4).  Find  the  traces  of  two  of  the  sides  of  the  triangle  and 
pass  the  plane  \]  perpendicular  to  the  plane  of  the  triangle  and 
perpendicular  to  f\,  and  through  the  point  D  (0,7,0). 


132  Engineering  Desceiptive  Geometry 

126.  Find  the  true  shape  of  the  triangle  A  (8,  6,1),  B  (4,  2,  9), 
C  (10,2,3).  Find  the  traces  of  two  of  the  sides  of  the  triangle 
and  pass  the  plane  U  perpendicular  to  the  plane  of  the  triangle 
and  perpendicular  to  H^  and  through  the  point  D  (0, 1,  0). 

127.  Draw  the  traces  of  a  plane  P  perpendicular  to  V  and  to 
the  line  A  (2,6,9),  B  (8,6,5)  at  C  (11,6,3).  If  this  plane  is 
used  as  an  auxiliary  plane  of  projection,  what  is  the  projection  of 
A^onit? 

128.  Draw  the  traces  of  a  plane  P  perpendicular  to  lil  and  ta 
the  line  A  (3,  9,  6),  B  (13,  4,  6),  at  C  (17,  2,  6),  a  point  on  AB. 
(If  wire-mesh  cage  is  used  for  the  solution,  turn  §'  to  serve  as  \J 
and  draw  on  it  the  view  of  AuBu.) 

129.  Draw  the  three  traces  of  a  plane  P  perpendicular  to  the 
oblique  line  A  (8, 12,  5),  5  (14,  3,  7).  Show  that  all  three  traces 
are  perpendicular  to  the  corresponding  projections  of  AB. 


CHAPTEE  XIII. 
THE  ELEMENTS  OF  ISOMETRIC  SKETCHING. 

119.  Isometric  Projection. — There  is  one  special  brancli  of 
Orthographic  Projection  which  is  of  peculiar  value  for  represent- 
ing forms  which  consist  wholly  or  mainly  of  plane  faces  at  right 
angles  to  each  other.  Ordinary  orthographic  views  are  projec- 
tions upon  planes  parallel  to  the  principal  plane  faces  of  the  object, 
as  shown  in  Fig.  2,  Art.  4.  If,  however,  instead  of  the  regular 
planes  of  projection,  the  object  is  projected  upon  a  new  plane  of 
projection,  making  the  same  angle  ivith  each  of  the  regular  planes, 
an  entirely  different  result  is  obtained,  called  an  '^isometric  pro- 
jection." This  view  has  the  useful  property  that  it  has  all  the  air 
of  a  perspective  and  ma}^,  witli  certain  restrictions,  be  used  alone 
without  other  views  as  a  full  representation  of  the  object. 

In  Art.  21  the  method  of  converting  the  perspective  drawings 
of  this  treatise  into  isometric  sketches  was  explained  in  a  rough 
and  unscientific  way.  In  this  chapter  there  is  explained  the  method 
of  making  isometric  sketches  from  models,  as  a  step  to  making 
orthographic  drawings  or  isometric  drawings. 

120.  Isometric  Sketches  of  Rectangular  Objects. — Figs.  19  and 
19a  are  the  isometric  drawings  of  a  cube.  Since  the  line  of  sight 
from  the  eye  to  the  point  0  makes  equal  angles  with  fil,  V  and  S, 
the  three  planes  must  subtend  the  same  angle  at  0.  XOY,  YOZ 
and  XOZ  are  each  120°.  though  representing  angles  of  90°  on  the 
cube.  Since  opposite  edges  of  H  are  parallel,  it  follows  that  each 
face  of  the  cube  is  a  rhombus  and  that  the  cube  appears  as  a  regular 
hexagon,  all  edges  appearing  of  exactly  the  same  length.  This 
fact  is  the  basis  of  the  name  "isometric,"  meaning  "equal- 
measured." 

Figs.  115,  IIG  and  117  are  sketches  of  other  objects,  all  of  whose 
corners  are  right  angles.    The  angles  at  these  comers  appear  there- 
10 


134 


Engineering  Descriptive  Geometry 


fore  like  those  of  the  cube,  either  as  60°  or  120°  on  the  isometric 
sketch. 

In  making  the  isometric  sketch  from  a  model  having  rectangular 
faces,  the  tirst  step  is  to  put  the  object  approximately  in  the  iso- 


Brick 
Fig.  115. 


Half  Joinf 

FiG.  116. 


Mortise 5* Tenon  Joint 
Fig.  117. 


Position  •for 
Orthographic 
Projection  . 

Fig.  118. 


Turned  45 
about  a  verti- 
cal  dxis. 

Fig.  119. 


Tilt-ed   35"-44-' 
about  an  hori- 
zontal   o?<.is. 

Fig.  120. 


metric  position.  At  any  projecting  corner  imagine  a  line  to  project 
from  the  corner  so  as  to  make  equal  angles  with  the  three  edges 
which  meet  at  the  given  corner.  View  the  object  by  sighting  along 
this  imaginary  line  and  begin  the  sketch  from  that  view. 


The  Elements  of  Isometric  Sketching  135 

If  there  is  any  difficulty  in  finding  this  line  of  vision  directly, 
the  object  may  be  turned  horizontally  through  an  angle  of  45  °  and 
tilted  down  through  an  angle  of  35°  44'.  This  operation  is  the 
basis  of  the  method  of  finding  the  "  isometric  projection." 

Figs.  118,  119  and  120  show  the  steps  in  passing  from  the  ortho- 
graphic position  to  the  isometric  position,  the  model  used  being  a 
rectangular  block  with, a  lengthwise  groove  cut  in  one  face. 

121.  Isometric  Axes. — It  will  be  noticed  in  the  previous  iso- 
metric figures  that  all  lines  are  drawn  in  one  of  three  general  direc- 
tions. One  of  these  directions  is  usually  taken  as  vertical  and  the 
other  two  directions  make  angles  of  120°  with  the  vertical.  These 
three  directions  are  known  as  the  isometric  axes.  In  this  sense 
the  word  axis  means  a  direction,  not  a  line. 

In  plotting  points  from  a  selected  origin,  the  x  coordinates  are 
plotted  up  and  to  the  left,  the  y  coordinates  up  and  to  the  right, 
and  the  z  coordinates  vertically  downward,  as  in  Fig.  19a. 

122.  Isometric  Paper. — Paper  ruled  in  the  direction  of  the  iso- 
metric axes  is  called  isometric  paper,  and  is  of  great  assistance  in 
making  isometric  sketches.  The  lines  divide  the  paper  into  small 
equilateral  triangles. 

In  sketching,  the  sides  of  these  equilateral  triangles  are  taken  to 
represent  unit  distances,  exactly  or  at  least  approximately.  Thus, 
if  the  model  shown  in  Fig.  120  is  a  block  3"  x  3"  X  8",  with  a  2"  x  1" 
groove  lengthwise  along  one  face,  some  point  a  on  the  paper  is 
selected,  and  from  it  distances  are  taken  along  the  isometric  axes, 
so  that  each  unit  space  represents  one  inch. 

From  a  three  units  are  counted  vertically  downward,  eight  up, 
and  to  the  right,  and  one  unit,  follov^ed  by  a  gap  in  the,  line  of  one 
unit,  and  then  a  second  unit,  up  to  the  left.  Thus  all  lines  of  the 
sketch  follow  the  ruled  lines  as  long  as  the  dimensions  of  the  model 
are  in  even  inches. 

An  isometric  sketch  made  in  this  manner,  particularly  if  spaces 
have  been  exactly  counted  off  according  to  the  dimensions  of  the 
piece,  is  practically  an  isometric  drawing.  If  fully  dimensioned,  a 
sketch  on  plain  paper  proportioned  by  the  eye  is  nearly  as  good  as 
one  in  which  spaces  are  counted  exactly.     Such  sketches  serve  all 


136 


Engineering  Descriptive  Geometry 


purposes,  though  of  course  more  difficult  to  make  than  those  on 
isometric  paper. 


Fig.  121. 


Fig.  122. 


123.  Non-Isometric  Lines  in  Isometric  Sketching.  —  Objects 
which  have  a  few  faces  and  edges  oblique  to  the  principal  plane 
faces  may  still  be  shown  by  isometric  sketching.  In  such  cases  it  is 
always  well  to  circumscribe  a  set  of  rectangular  planes  about  the 


Fig.  123. 


oblique  parts  of  the  object  to  aid  the  imagination.  Dimension 
extension  lines  should  be  used  for  this  purpose.  In  using  isometric 
paper  this  squaring  up  is  done  by  the  lines  of  the  paper. 


The  Elements  of  Isometric  Sketching 


137 


Figs.  122,  123  and  124  are  good  examples  of  oblique  lines  and 
faces.  Figs.  123  and  124  show  also  the  circumscribed  isometric 
lines  which  "  square  up ''  the  oblique  parts. 

124.  Angles  in  Isometric  Sketching. — In  isometric  sketching 
angles  do  not,  as  a  rule,  appear  of  their  true  magnitude.  Thus  the 
90°  angles  on  the  faces  of  the  brick  appear  in  Fig.  115  as  60°  or 
120°,  but  not  as  90°.  In  general,  the  lengths  of  oblique  or  inclined 
lines  depend  on  position,  and  are  not  subject  to  measurement  by- 
scale. 

The  lines  which  square  up  oblique  parts  are  useful  in  giving  tSe 
tangent  of  the  angle  of  an  oblique  surface.  Thus  in  Fig.  124,  the 
angle  a  differs  in  reality  from  the  angle  as  it  appears  in  either  place 

marked,  but  the  tangent  of  a  is  -!i  .     In  Fig.  123,  0=tsin-^  VI .     In 

2^  n 

practice  angles  are  often  given  by  their  tangents.    Thus  the  slope  of 


Fig.  126. 


Fig.  127. 


a  roof  is  given  as  "  one  in  two  "  or  the  gradient  of  a  railroad  as 
"  three  per  cent.'' 

125.  Cylindrical  Surfaces  in  Isometric  Sketching. — In  ortho- 
graphic drawings  circles  appear  commonly  on  planes  parallel  to  the 
three  planes  of  projection.  To  illustrate  the  position  and  appear- 
ance of  circles  in  isometric  drawing  in  the  three  typical  cases.  Fig. 
125  represents  the  isometric  sketch  of  a  cube,  having  a  circle  in- 
scribed in  each  square  face. 

Each  of  the  faces  of  the  cube  is  perpendicular  to  the  isometric 
axis  given  by  the  intersection  of  the  other  two  faces.  Thus  the 
square  ABCD  is  perpendicular  to  the  edge  BF.     The  circle  ahcd. 


138  Engineering  Descriptive  Geometry 

inscribed  in  the  square  ABCD,  appears  as  an  ellipse,  whose  minor 
axis,  ef,  lies  on  the  diagonal  BD  of  the  square,  BD  appearing  as  a 
continuation  of  the  edge  FB,  In  all  three  cases,  then,  the  minor 
axis  of  the  ellipse  lies  in  the  same  direction,  on  the  sketch,  as  that 
isometric  axis  to  which  the  plane  of  the  circle  is  in  reality  perpen- 
dicular. 

The  major  axis  is  necessarily  perpendicular  to  the  minor  axis, 
and  lies  on  the  other  diagonal  of  the  square. 

Since  the  cylinder  is  the  curved  surface  most  used  in  engineering, 
the  rule  may  be  applied  to  cylinders  as  follows :  The  ellipse  which 
represents  the  circular  base  of  any  cylinder  must  be  so  sketched 
that  its  minor  axis  is  in  line  with  the  axis  or  center  line  of  the 
cylinder.  Fig.  126  is  an  isometric  sketch  of  a  piece  composed  of 
cylinders.    All  the  ellipses  are  seen  to  follow  this  rule. 

In  sketching  cylindrical  parts  of  objects,  it  is  necessary  to  im- 
agine them  squared  up  by  the  use  of  isometric  lines  and  planes. 
Thus  the  first  steps  in  sketching  the  piece  of  Fig.  126  are  shown 
in  Fig.  127.  The  circumscribing  of  a  square  about  a  circle  in  the 
object  corresponds  to  circumscribing  a  rhombus  about  the  ellipse 
in  the  isometric  sketch.  It  now  remains  to  inscribe  an  ellipse  in 
the  rhombus.  This  ellipse  must  be  tangent  to  the  rhombus  at  the 
middle  of  each  side.  To  sketch  the  ellipse,  as  for  example  the  small 
end  in  Fig.  127,  draw  the  diagonals  of  the  rhombus  to  get  the 
directions  of  the  major  and  minor  axes,  and  find  the  middle  points 
of  the  sides  (by  center  lines,  through  the  intersection  of  the  diagon- 
als). It  is  now  easy  to  sketch  the  ellipse,  having  four  points  given, 
the  direction  of  passing  through  those  points,  and  the  directions  of 
the  major  and  minor  axes. 

126.  Isometric  Sketches  from  Orthographic  Sketches. — A  good 
exercise  consists  in  making  isometric  sketches  from  orthographic 
sketches  or  drawings.  The  three  coordinate  directions,  x,  y  and  z, 
must  be  kept  in  mind  at  all  times.  Fig.  128,  as  an  example,  is  most 
instructive.  From  the  orthographic  sketches,  Fig.  128,  the  iso- 
metric sketch.  Fig.  129,  is  to  be  made.  A  point  a  is  selected  to  rep- 
resent a  point  a  on  the  orthographic  views.  The  line  ah  is  an  x 
dimension  and  is  plotted  up  to  the  left ;  ac  is  a  ?/  dimension,  and  is 
plotted  up  to  the  right;  while  a.^  is  a  2  dimension,  and  is  plotted 


The  Elements  of  Isometric  Sketching 


139 


vertically  downward.  The  semicircle  is  inscribed  in  a  half -rhombus, 
tangent  at  b,  e  and  /. 


— h  1  1  1  1 ^^^><^<^^l^s>^^H^<'^<' 

J  >\  ^<^  S<r  ^^S<r  ^  cT  *^ccr^^  ^>N  ^^  ^<^ 

'    ^sC^    ^^'^^^^'^^>vT>c  >c^'^^'^' 

-^ /f\ P% x"vl^ S ^  ^  ^'NaTS'^  "^^^k^^-^^  ^'' 

^^^2^^sx"\^^  ^<^pk^  x\D^^!^  V  ^' 

L          C^  ^^  ^<C  /  C  !^^^  \  ^'^j^]^^^lS^  S^^' 

£          lb        ■^'^L^s^  ^^  v^''^<^\<'^iN^s 

(L  ^\^\ ^ ^ ^^>i^ >^^iiyf*^'\-^^-^ ^ /x 

^                  a_               /'^  2^!  ^  \ /'>;>^^  /  "S^  Q^^^'\/>\  ^  C  " 

r~          c   ~^^^^^^ ^ '^ ">^$ ^^< ^  <i^   S<' 

^/\  ^\  ^  Vj^  \^  ><  s  ^1^  !^^4^D^  ^^^\  /  \  !^  * 

V                 /             ^  ^^  /  \  ^^^"v  !i^^!^^  !!^4\  ^  V  !^^^  \  ^ 

V,      V          ^  3^  3^  "^^  ^ sic  ^\  s^kr  ^ ^ "^^^  ^   ^ 

^^^^  <p   /T^S^^^  *^   S  5^$ 

— -.-     ^^^^'^^-^^c^'^^^c '^'^ ^ '^^'^'^'^ > 

:i.^\^^^^\^\!^x!^\Ss^V^\!^\!^\ 

x^<^Sc  ^<r  ^c  ^c  ^<r^^  ^  ^  ^cT  >cr  ^<' 

><J^<i><^><^>x>x>xPK5\>x>x 

Fig.  128. 


Fig.  129. 


The  cross-section  lines  of  Fig.  128  and  the  isometric  lines  of  Fig. 
129  are  represented  as  overlapping  between  the  figures.  Some  iso- 
metric paper  is  ruled  in  this  manner,  so  that  it  may  be  used  for 
both  purposes. 


140 


Engineering  Descriptive  Geometry 


Problems  XIII. 

(For  blackboard  or  isometric  paper.) 

130.  Make  an  isometric  sketch  of  the  angle  piece,  Fig.  130,  using 
the  spaces  for  1"  distances. 


Fig.  130. 


Fig.  131. 


131.  Measure  the  tool-chest,  Fig.  131,  scale,  5"=1  foot,  and 
make  a  bill  of  material,  tabulating  the  boards  used,  and  recording 
their  sizes,  giving  dimensions  in  the  order :  width,  thickness,  length, 
thus : 

Mark.  Name.  Size.  Number. 

A.  Top  of  Chest.  14"  x   1"  x   24".  2. 

132.  A  parallelopiped,  9"  x  6"  x  3",  has  a  3"  square  hole  from 
center  to  center  of  the  largest  faces,  and  a  2"  bore-hole  centrally 
from  end  to  end.    Make  an  isometric  sketch. 

133.  Let  Fig.  3,  Art.  5,  represent  a  model  cut  from  a  12"  cube 
by  removing  the  center,  leaving  the  thickness  of  the  walls  3".  Let 
the  angular  point  form  a  triangle  whose  base  is  12"  and  altitude  8". 
Make  an  isometric  sketch. 


The  Elements  of  Isometric  Sketching  141 

134.  A  cube  of  10"  has  a  6"  square  hole  piercing  it  centrally  from 
cne  side  to  the  other,  and  a  4"  bore-hole  piercing  it  centrally  from 
side  to  side  at  right  angles  to  the  larger  hole.  Make  an  isometric 
sketch. 

135.  A  grating  is  made  by  nailing  slats  f"xi"xl2",  spaced  J" 
apart,  on  three  square  pieces,  IJ"  square,  22"  long,  spaced  4^"  apart. 
Make  an  isometric  sketch. 

136.  Make  orthographic  sketches  of  the  bracket.  Fig.  122.  Views 
required  are  plan  and  front  elevation.  (On  cross-section  paper  use 
the  unit  distance  for  the  unit  of  the  isometric  paper.  On  black- 
board let  each  unit  of  the  isometric  paper  be  represented  by  a  dis- 
tance of  2".) 

137.  Make  isometric  sketches  of  Fig.  11,  Art.  14,  and  Fig.  24, 
Art.  32. 

138.  Make  isometric  sketches  of  Fig.  13,  Art.  15,  and  of  Fig.  82, 
Art.  89.  In  Fig.  82  let  A  be  the  point  (9,  8,  0)  and  B  the  point 
(9,0,12). 

139.  Make  an  isometric  sketch  of  Fig.  71,  Art.  80,  the  diameter 
of  the  cylinder  being  7  units  and  the  length  14  units. 

140.  Make  an  isometric  sketch  of  Fig.  92,  Art.  93,  using  the 
coordinates  given  in  Art.  94. 


CHAPTER  XIV. 
ISOMETRIC  DRAWING  AS  AN  EXACT  SYSTEM. 

127.  The  Isometric  Projection  on  an  Oblique  Auxiliary  Plane. — 

The  sketches  previously  considered  have  generally  had  no  exact  scale. 
Those  drawn  on  isometric  paper  have  a  certain  scale  according  to 
the  distance  which  one  unit  space  of  the  paper  actually  represents. 


If  the  isometric  projection  is  derived  from  an  orthographic  draw- 
ing of  the  usual  kind  by  the  laws  of  projection,  the  isometric  projec- 
tion so  formed  has  of  course  the  same  scale  as  the  original  drawing. 

In  Fig.  132  an  isometric  projection  of  a  cube  is  derived  from  the 
orthographic  drawing  by  the  use  of  an  inclined  plane  of  projection. 


Isometric  Drawing  as  an  Exact  System  143 

HJ,  and  an  oblique  auxiliary  plane  of  projection  "W-  The  aim  is 
to  produce  the  projection  on  a  plane  making  the  same  angle  with  all 
three  edges  of  the  cube  meeting  at  any  one  corner.  This  plane  must 
be  perpendicular  to  a  diagonal  of  the  cube.  In  Fig.  132  this  di- 
agonal is  the  line  EC,  a.  true  diagonal,  passing  through  the  center  of 
the  cube,  not  a  diagonal  of  one  face  of  the  cube. 

The  first,  or  inclined,  auxiliary  plane  U  is  taken  parallel  to  the 
V  projection  of  EC,  and  therefore  perpendicular  to  V  and  making 
an  angle  of  45°  with  H  and  S-  The  projection  of  EC  on  U  shows 
its  true  length. 

The  second,  or  oblique,  auxiliary  plane  'W  is  taken  perpendicular 
to  EC.  It  is  oblique  as  regards  H  and  V?  b^t,  as  EC  is  a  line  par- 
allel to  U,  and  W  is  perpendicular  to  EC,  W  is  perpendicular  to 
U.  As  regards  V  and  U?  W  is  an  inclined  plane,  having  its  in- 
clined trace  MN  on  U^  the  trace  on  V  being  a  line  MLv,  perpendic- 
ular to  ZM,  the  trace  of  U  on  V-  The  construction  of  this  second 
projection  is  therefore  according  to  the  usual  methods.  Any  point, 
as  F,  is  projected  by  a  perpendicular  line  across  the  trace  MN  and 
the  distance  nFw  is  laid  off  equal  to  mFv. 

The  projection  on  'W  is  the  isometric  projection  of  the  cube  and 
is  full-size  if  the  plan  and  front  elevation  are  full-size  projections. 
The  edges  are  all  foreshortened,  however,  and  measure  only  j-i^\  of 
their  true  length. 

128.  The  Angfles  of  the  Auxiliary  Planes. — The  plane  U  makes 
an  angle  of  45°  with  the  plane  ff.  The  plane  W  makes  an  angle 
of  35°  44'  with  g,  or  (90° -35°  44')  with  V-  If  the  side  of  the 
cube  is  taken  as  1,  the  length  of  the  diagonal  of  the  face  of  the  cube 
is  V2,  and  the  length  of  the  diagonal  of  the  cube  is  \^.     The 

first  angle  is  that  angle  whose  tangent  is  -  ^  ,  or  whose  sine  is  ~7^* 
The  second  angle  is  that  angle  whose  tangent  is  -— ^  and  whose  sine 

129.  The  Isometric  Projection  by  Rotating  the  Object. — In  Fig. 
134  is  shown  a  method  of  deriving  the  isometric  projection  by  turn- 
ing the  object.    The  plan,  front,  and  side  elevations  are  drawn  with 


144 


Engineering  Descriptive  Geometry 


the  object  turned  through  an  angle  of  45°  from  the  natural  posi- 
tion (that  in  which  the  faces  of  the  cube  are  all  parallel  to  the 
reference  planes) .  The  side  elevation  shows  the  true  length  of  one 
diagonal  of  the  cube,  AG.  Some  point  on  AG  extended,  as  K,  is 
taken  as  a  pivot,  and  the  whole  object  is  tilted  down  through  an 
angle  of  35°  44',  bringing  AG  into  a  horizontal  position,  A'G'.    The 


new  projection  of  the  object  in  V  is  the  isometric  projection.  This 
process  of  turning  the  object  corresponds  to  the  turning  of  the 
object  in  isometric  sketching,  as  shown  in  Figs.  118,  119  and  120. 

The  isometric  projection  of  the  cube  has  all  eight  edges  of  the 
same  length,  but.  foreshortened  from  the  true  length  in  the  ratio  of 
V3  to  V2. 

Any  object  of  a  rectangular  nature  may  be  treated  by  either 
process  to  obtain  the  isometric  projection. 


Isometric  Drawing  as  an  Exact  System  145 

130.  The  Isometric  Drawing. — To  make  a  practical  system  of 
drawing  capable  of  representing  rectangular  objects  in  an  unmis- 
takable manner  in  one  view,  the  fact  that  all  edges  are  foreshort- 
ened alike  is  seized  upon,  but  the  disagreeable  ratio  of  foreshorten- 
ing is  obviated  by  ignoring  foreshortening  altogether. 

An  isometric  drawing  is  one  constructed  as  follows:  On  three 
lines  of  direction,  called  isometric  axes,  making  angles  of  120°  with 
each  other,  the  true  lengths  of  the  edges  of  the  object  are  laid  off. 
These  lengths,  however,  are  only  those  which  are  mutually  at  right 
angles  on  the  object.  All  otlier  lines  are  altered  in  shape  or  length. 
An  isometric  drawing  is  distinct  from  an  isometric  projection,  as 
it  is  larger  in  the  proportion  of  100  to  83  (V3:  V2).  The  iso- 
metric drawing  of  a  1"  cube  is  a  hexagon  measuring  1"  on  each 
edge. 

131..  Requirement  of  Perpendicular  Faces. — An  isometric  draw- 
ing, being  a  single  view,  cannot  really  give  "  depth,"  or  tell  exactly 
the  relative  distances  of  different  points  of  the  object  from  the  eye. 
It  absolutely  requires  that  the  object  drawn  shall  have  its  most 
prominent  faces,  at  least,  mutually  perpendicular.  The  mind  must 
be  able  to  assume  that  the  object  represented  is  of  this  kind,  or  the 
drawing  will  not  be  "  read "  correctly.  Even  on  this  assumption, 
in  some  cases  isometric  drawing  of  rectangular  objects  may  be 
misunderstood  if  some  projecting  angle  is  taken  as  a  reentrant  one. 
Thus  in  Fig.  133  we  have  a  drawing  which  might  be  taken  as  the 
pattern  of  inlaid  paving  or  other  flat  object.  If  it  is  taken  as  an 
isometric  drawing  and  the  various  faces  are  assumed  to  be  perpen- 
dicular to  each  other,  it  becomes  the  drawing  of  a  set  of  cubes. 
Curiously  enough,  it  can  be  taken  to  represent  either  6  or  7  cubes, 
according  as  the  point  A  is  taken  as  a  raised  point  or  as  a  depressed 
one.  In  other  words,  it  even  requires  one  to  know  just  how  the 
faces  are  perpendicular  to  each  other  to  be  able  to  take  the  drawing 
in  the  way  intended. 

This  requirement  of  perpendicular  faces  limits  the  system  of 
drawing  to  one  class  of  objects,  but  for  that  class  it  is  a  very  easy, 
direct,  and  readily  understood  method.  Untrained  mechanics  can 
follow  isometric  drawings  more  easily  than  orthographic  drawings. 


146 


Engineering  Descriptive  Geometry 


IsoMETKTC  Drawing  as  an  Exact  System  147 

132.  The  Representation  of  the  Circle. — In  executing  isometric 
drawings,  the  circle,  projected  as  an  ellipse,  is  the  one  drawback  to 
the  system.  To  minimize  the  labor,  an  approximate  ellipse  must 
be  substituted  for  an  exact  one,  even  at  the  expense  of  displeasing 
a  critical  eye.  The  system,  if  used,  is  used  for  practical  purposes 
where  beauty  must  be  sacrificed  to  speed.  In  Fig.  125  the  rhombus 
ABCD  is  the  typical  rhombus  in  which  the  ellipse  must  be  inscribed. 
The  exact  method  is  shown  in  Fig.  43,  but  requires  too  much  time 
for  constant  use.  The  following  draftsman's  ellipse,  devised  to  be 
exactly  tangent  to  the  rhombus  at  the  middle  point  of  each  side,  is 
reasonably  accurate.  From  B,  one  extremity  of  the  short  diagonal 
of  the  rhombus,  drop  perpendiculars  Bd  and  Be  upon  opposite 
sides,  cutting  the  long  diagonal  at  k  and  I.  With  5  as  a  center  and 
Bd  as  a  radius,  describe  the  arc  dc.  Similarly,  with  D  as  a  center, 
describe  the  arc  ha.  With  Tc  and  I  as  centers,  and  hd  as  a  radius, 
describe  the  arcs  ad  and  ch.  The  resulting  oval  has  the  correct 
major  axis  within  one-eighth  of  1  per  cent,  and  has  the  correct 
minor  axis  within  3^  per  cent. 

This  draftsman's  ellipse  is  exact  where  required,  namely,  on  the 
two  diameters  ac  and  dh,  which  are  isometric  axes,  and  it  is  prac- 
tically exact  at  the  extremity  of  the  major  axis. 

133.  Set  of  Isometric  Sketches. — Fig.  135  is  a  set  of  isometric 
sketches  of  the  details  of  the  strap  end  of  a  small  connecting-rod, 
from  which  to  make  orthographic  drawings.  The  isometric  sketch 
is  much  clearer  than  the  corresponding  orthographic  sketch,  and 
the  set  shows  clearly  how  the  pieces  are  assembled. 

The  orthographic  drawing  of  the  assembled  rod  end  is  much 
easier  to  make  than  the  assembled  isometric  drawing.  It  is  in  fact 
clearer  for  the  mechanic  than  the  assembled  isometric  drawing 
would  be,  for  the  number  of  lines  would  in  that  case  be  quite  con- 
fusing. It  illustrates  well  the  fact  that  isometric  sketches  and 
drawings  should  be  limited  to  fairly  simple  objects. 

Another  noteworthy  fact  is  that  center  lines,  which  should  always 
mark  s}Tnmetrical  parts  in  orthographic  drawings,  should  be  used 
in  isometric  drawing  only  when  measurements  are  recorded  from 
them. 

The  sketch  as  given  is  taken  directly  from  an  examination  paper 
used  at  the  U  S.  Naval  Academy  for  a  two-hour  examination.    On 


148  Engineering  Descriptive  Geometry 

account  of  the  shortness  of  the  period,  only  one  orthographic  view, 
the  front  elevation,  is  required,  but  if  time  were  not  limited,  a  plan 
also  should  be  drawn. 

The  following  explanation  of  the  sheet  is  printed  on  the  original : 
''Explanation  of  Mechanism. — ^The  isometric  sketches  represent 
the  parts  of  the  strap  end  of  a  connecting-rod  for  a  small  engine. 
In  assembling.  A,  B,  C,  and  D  are  pushed  together,  with  the  thin 
metal  liners,  G,  filling  the  space  between  B  and  C.  The  tapered 
key,  E,  is  driven  in  the  J"  holes  of  A  and  D,  which  will  be  found  to 
be  in  line,  except  for  a  displacement  of  ^"  which  prevents  the  key 
from  being  driven  down  flush  with  the  top  of  the  strap  D.  The  two 
bolts,  F,  are  inserted  in  their  holes,  nuts  H  screwed  on,  and  split 
pins  (which  are  not  drawn)  inserted  in  the  -J"  holes,  locking  the 
nuts  in  place.  In  time  the  bore  of  the  brasses  B  and  C  wears  to 
oval  form.  To  restore  to  circular  form,  one  or  two  liners  would  be 
removed  and  the  strap  replaced.  The  key  driven  in  would  then 
draw  the  parts  closer  by  the  thickness  of  the  liners  removed. 

"Drawing  (to  he  Orthographic,  not  Isometric). — On  a  sheet 
14"xll"  make  in  ink  a  working  drawing  of  the  front  elevation  of 
the  rod  end  assembled,  viewed  in  the  direction  of  the  arrow.  Put 
paper  with  long  dimension  horizontal.  Put  center  of  bore  of 
brasses  4"  from  left  edge  of  paper  and  5"  from  top  edge.  No 
sketch,  no  legend,  no  dimensions.'' 

Problems  XIV. 

140.  An  ordinary  brick  measures  S"x4^"x2^".  Make  an  ortho- 
graphic drawing  and  an  isometric  projection  after  the  manner  of 
Fig.  132,  Art.  127.  Contrast  it  with  the  isometric  drawing  made 
according  to  Art.  130. 

.  141.  Make  the  isometric  projection  of  the  brick,  8"x4"x2^", 
turning  it  through  the  angles  of  45°  and  35°  44',  as  in  Fig.  134, 
Art.  129. 

142.  From  Fig.  135  make  a  plan  and  front  elevation  of  the 
strap  D. 

143.  From  Fig.  135' make  a  plan  and  front  elevation  of  the  stub 
end  A. 

144.  From  Fig.  135  make  a  plan  and  front  elevation  of  the 
brass  C. 


SET  OF  DESCRIPTIVE  DRAWINGS. 

The  following  four  drawing  sheets  are  designed  to  be  executed  in 
the  drawing  room  to  illustrate  those  principles  of  Descriptive 
Geometry  which  have  the  most  freouent  application  in  Mechanical 
or  Engineering  Drawing. 

The  paper  used  should  be  about  28"x22",  the  drawing-board  of 
the  same  size,  and  the  blade  of  the  T-square  30". 

To  lay  out  the  sheets  find  the  center,  approximately,  draw  center 
lines,  and  draw  three  concentric  rectangles,  measuring  24"  x  18", 
22"xl6",  and  21"xl5".  The  outer  rectangle  is  the  cutting  line 
to  which  the  sheets  are  to  be  trimmed.  The  second  one  is  to  be  inked 
for  the  border  line.  The  inner  one  is  described  in  pencil  only  as  a 
"working  line,"  or  line  outside  of  which  no  part  of  the  actual 
figures  should  extend.  The  center  lines  and  other  fine  lines,  in- 
cluding dimensions,  may  extend  beyond  the  working  line.  In  the 
lower  right  corner  reserve  a  rectangle  6"  x  3",  touching  the  working 
lines,  for  the  legend  of  the  drawing. 

In  making  the  drawings  three  widths  of  line  are  used. 

The  actual  lines  of  the  figures  must  be  "  standard  lines  "  or  lines 
not  quite  one-hundredth  of  an  inch  thick.  The  thin  metal  erasing 
shield  may  be  used  as  a  gauge  for  setting  the  right-line  pen,  by  so 
adjusting  the  pen  that  the  shield  will  slowly  slip  from  between  the 
nibs,  when  inserted  and  allowed  to  hang  vertically.  Visible  edges 
are  full  lines.  Hidden  edges  are  broken  lines;  the  dashes  J"  long 
and  spaces  ^y"  long. 

The  extra-fine  lines  ave  described  with  the  pen  adjusted  to  as  fine 
a  line  as  it  will  carry  continuously.  The  axes  of  projection  are 
fine  full  lines.  The  dimension  lines  are  long  dashes,  J"  to  1"  long, 
with  -J"  spaces.  The  center  lines  are  long  dashes  with  fine  dots 
between  the  dashes,  or  are  dash-dot  lines.  The  construction  lines 
are  long  dashes  wdth  two  dots  between,  or  are  dash-dot-dot  lines. 
When  auxiliar}^  cutting  planes  are  used,  one  only,  together  with  its 
corresponding  projection  lines,  should  be  inked  in  this  manner. 
11 


150  Engineering  Descriptive  Geometry 

The  extra-heavy  lines  are  about  two-hundredths  of  an  inch  thick, 
and  are  for  two  purposes :  for  shade  lines,  if  used ;  and  for  paths  of 
sections,  or  lines  showing  where  sections  have  been  taken,  as  pq. 
Fig.  32.  These  paths  of  sections  should  be  formed  of  dashes 
about  J"  long. 

SHEET  I:     PRISMS  AND  PYRAMIDS. 

Lay  out  the  sheet  and  from  the  center  of  the  sheet  plot  three  ori- 
gins :  The  first  origin  5^"  to  the  left  and  4^"  above  the  center  of  the 
sheet;  the  second  8"  to  the  right  and  2-i"  above  the  center;  and  the 
third  A:"  to  the  left  and  4V'  below  the  center.  Pass  vertical  and 
horizontal  lines  through  these  points  to  act  as  axes  of  projection. 

First  Origin:     Pentagonal  Prism  and  Inclined  Plane. 

Describe  a  pentagonal  prism,  the  axis  extending  from  P  (2% 
IJ",  \")  to  P'  (2",  IJ",  2f' )•  The  top  base  is  a  regular  pentagon 
inscribed  in  a  circle  of  IJ"  radius,  one  corner  of  the  pentagon 
being  at  A  (2",  J",  \").  Draw  three  views  of  the  prism.  Draw  the 
traces  of  a  plane  P,  perpendicular  to  V>  its  trace  on  V  passing 
through  the  point  c  (0",  0",  24")  and  making  an  angle  of  60°  with, 
the  axis  of  Z.  Draw  on  the  side  elevation  the  line  of  intersection 
of  the  prism  and  the  plane  P.  Show  the  true  shape  of  the  polygonal 
line  of  intersection  on  an  auxiliary  plane  KJ?  perpendicular  to  V> 
its  traces  on  V  passing  through  the  point  (0",  0",  4^").  On  JJ 
show  only  the  section  cut  by  the  plane.  Draw  the  development 
of  the  surface  of  the  prism,  with  the  line  of  intersection  described 
on  it.  Draw  the  left  edge  of  the  development  [representing' 
A  (2",  i",  I"),  A'  (2'Vi",  2Y)^  as  a  vertical  line  V  to  the  right  of 
the  axis  of  Y,  and  use  the  top  working  edge  of  the  sheet  as  the  top 
line  of  the  development.     Omit  the  pentagonal  bases. 

Second  Origin:    Octagonal  Prism  and  Triangular  Prism. 

Describe  an  octagonal  prism,  the  axis  extending  from  P  (2^, 
If"  i")  to  P'  (21",  If",  4i").  The  octagonal  base  is  circumscribed 
about  a  circle  of  2J''  diameter,  one  flat  side  being  parallel  to  the 


Set  of  Descriptive  Drawings  151 

axis  of  X.  Describe  a  triangular  prism,  its  axis  extending  from 
Q  (0r52,  If",  li")  to  Q'  (3'.'98,  If",  3^"),  intersecting  TF'  at  its 
middle  point  and  making  an  angle  of  60°  with  it.  The  base  is  in 
a  plane  perpendicular  to  QQ\  and  is  an  equilateral  triangle  cir- 
cumscribed about  a  circle  of  \"  diameter.  One  corner  is  at 
J  (1",  If",  0".38).  Draw  the  H,  V,  and  S  projections  of  the 
prisms  and  a  complete  projection  on  a  plane  JJ?  taken  perpendicular 
to  QQ' ,  and  whose  trace  on  V  passes  through  the  point  \^" ,  0",  0"). 
'Draw  the  triangular  prism  as  if  piercing  the  octagonal  prism. 

Third  Origin:    Hexagonal  Pyramid  and  Square  Prism. 

Describe  an  hexagonal  pyramid,  vertex  at  P  (IJ"?  2",  J"),  center 
of  base  at  F'  (IJ",  2",  3") .  The  hexagonal  base  is  in  a  plane  parallel 
to  H  and  is  circumscribed  about  a  circle  2J"  in  diameter,  one 
corner  being  at  A  (If",  0'.'5G,  3").  Projecting  from  the  sides  of  the 
pyramid  are  two  portions  of  a  square  prism,  whose  axis  is  Q  (J", 
2'',  3i"),  Q'  (3i",  2'V2i").  The  square  base  is  in  a  plane  parallel 
to  S  and  measures  V  on  each  edge,  and  its  edges  are  parallel  to  the 
axes  of  Y  and  Z.  Letter  the  edges  GG\  HH',  etc.,  the  point  G 
being  (i",  IJ",  If"),  H  (i",  ^",  1%"),  etc.  Draw  the  object  as  if 
cut  from  one  solid  piece  of  material,  the  prism  not  piercing  the 
pyramid. 

The  views  required  are  plan,  front  elevation,  and  side  elevation, 
and  also  an  auxiliary  projection  on  a  plane  \],  perpendicular  to  W. 
The  IMI  trace  of  U  makes  an  angle  of  120°  with  the  axis  of  X  at 
the  point  Z  (2f",  0",  0"). 

Draw  also  the  developments  of  the  surfaces.  Place  the  vertex  of 
the  developed  pyramid  at  a  point  J"  to  the  right  and  3J"  above  the 
origin,  and  the  point  A  -J"  to  the  right  and  0."36  above  the  origin. 
Mark  the  line  of  intersection  with  the  prism  on  this  development. 

Between  the  side  elevation  and  the  legend  space,  draw  the  de- 
velopment of  the  square  prism,  placing  the  long  edges,  OG',  RE', 
etc.,  in  a  vertical  position.  Describe  the  line  of  intersection  on  the 
development.  Let  the  edge  which  has  been  opened  out  be  GG',  and 
let  the  middle  portion  of  the  prism,  which  does  not  in  reality  exist, 
be  drawn  with  construction  lines. 


152  Engineering  Descriptive  Geometry 

General  Directions  for  Completing  the  Sheet. 

In  inking  the  sheet  show  one  line  of  projection  for  the  determi- 
nation of  one  point  on  each  line  of  intersection.  Shade  the  figure, 
except  the  developments. 

In  the  legend  space  make  the  following  legend : 

SHEET    I.  (Block  letters  15/32"  high.) 

DESCRIPTIVE   geometry.  (All  caps  3/16"  high.) 

PRISMS   AND  PYRAMIDS.  (All  caps  9/32"  high.) 

Name    (signature).  ClasS.       (Caps  1/8"  high,  lower  case  1/12"  high.) 

Date.  (Caps  1/8"  high,  lower  case  1/12"  high.) 


SHEET  II:     CYLINDERS,  ETC. 

Lay  out  cutting,  border,  and  working  lines,  and  legend  space  as 
before. 

Plot  four  points  of  origin  as  follows :  First  origin,  6"  to  the  left 
and  4"  above  the  center  of  the  sheet ;  second  origin,  4 J"  to  the  right 
and  4J"  above  the  center;  third  origin,  6 J"  to  the  left  and  3 J" 
below  the  center;  fourth  origin,  6^  to  the  right  and  4 J"  below  the 
center. 

First  Origin:     Intersecting  Right  Cylinders. 

Draw  the  three  views  of  two  intersecting  right  cylinders.  The 
axis  of  one  is  P  (2J",  2",  Y),P'  m",  ^",  H"),  and^its  diameter  is 
3".  The  axis  of  the  other  is  Q  {I",  1%",  2"),  Q'  (4^",  If",  2"),  and 
its  diameter  2f".  Determine  the  line  of  intersection  in  V  by  planes 
parallel  to  V  at  distances  of  %",  1%  IJ",  etc. 

Second  Origin:     Inclined  Cylinder  and  Inclined  Plane. 

Draw  three  views  of  an  inclined  circular  cylinder,  cut  by  a  plane. 
The  axis  of  the  cylinder  is  P  (3.73",  If",  J"),  P'  (2",  If",  3i"). 
The  base  is  a  circle,  diameter  2^",  in  a  plane  parallel  to  H-  The 
plane  cutting  the  cylinder  is  perpendicular  to  V^  and  its  trace  in 
V  passes  through  the  middle  point  of  PP\  and  inclines  up  to  the 


Set  of  Descriptive  Drawings  153 

left  at  an  angle  of  30°  with  OX.  Plot  the  intersection  in  H,  V, 
and  S  and  find  the  true  shape  of  the  ellipse  by  an  auxiliary  plane 
of  projection  perpendicular  to  V  through  the  point  (3'',  0",  4"). 

Third  Origin:    Right  Circular  Cone  and  Inclined  Plane. 

Draw  a  right  circular  cone,  vertex  at  F  (2",  IJ",  ^'),  center  of 
base  at  F'  (2",  If",  4"),  diameter  of  base  3".  The  cone  is  inter- 
sected by  a  plane  perpendicular  to  §,  having  its  trace  in  §  parallel 
to  the  extreme  right  element  of  the  cone  and  through  the  point 
(0",  2^",  A"),  Draw  the  line  of  intersection  in  plan  and  front 
elevation,  and  show  the  true  shape  of  the  curve  by  projection  on  an 
auxiliary  plane  U  perpendicular  to  §,  its  trace  passing  through 
the  point  (0",  2^",  0"). 

Fourth  Origin:    Ogival  Point,  Vertical  Plane  and  Inclined  Plane. 

Let  S  li^  to  the  right  of  W  and  make  no  use  of  V-  The  problem 
is  to  draw  two  views  of  a  3^"  ogival  shell,  intersected  by  two  planes. 
The  ogival  point  is  generated  by  revolving  60°  of  arc  of  3 J"  radius 
about  an  axis  perpendicular  to  H  at  the  point  (2",  IJ",  0").  The 
initial  position  of  the  generating  arc  is  as  follows:  The  center  is 
at  D  (0",  3 J",  3J"),  one  extremity  is  at  B  (0",  0",  34"),  and  one  is  at 
F  (2",  IJ",  0.46").  The  cylindrical  body  of  the  shell  extends  from 
the  ogival  point  to  the  right  in  the  side  elevation,  a  distance  of  f ". 
Two  planes,  T  and  R,  intersect  the  shell.  T  is  parallel  to  and  1-J" 
from  §.  F  is  perpendicular  to  S,  and  its  S  trace  passes  through 
the  origin,  and  makes  angles  of  45°  with  the  axis  of  Y  and  the 
axis  of  Z.  Draw :  The  traces  of  T  and  F;  the  side  elevation ;  the 
line  of  intersection  of  T  with  the  shell;  and,  on  the  plan,  the  line 
of  intersection  of  F  with  the  shell. 

General  Directions  for  Completing  the  Sheet. 

■  In  inking  the  sheet  show  one  cutting  plane  for  the  determination 
of  each  line  of  intersection,  and  show  clearly  how  one  point  is  de- 
termined in  each  view  of  each  figure.  Shade  the  figure  except  the 
developments. 


154  EXGINEEEIXG    DESCRIPTIVE    GEOMETRY 

In  the  legend  space  make  the  following  legend : 

SHEET    II.  (Block  letters  15/32"  high.) 

DESCRIPTIVE  GEOMETRY.  (All  caps  3/16"  high.) 

INTERSECTIONS  OF   CYLINDERS,  ETC.  (AH  caps  9/32"  high.) 

Name    (signature).  ClaSS.  (Caps  1/8"  high,  lower  case  1/12"  high.) 

Date.  (Caps  1/8"  high,  lower  case  1/12"  high.) 


SHEET  III:  SURFACES  OF  REVOLUTION. 

Lay  out  center  lines,  cutting,  border  and  working  lines,  and 
legend  space  as  before. 

Plot  five  points  of  origin  as  follow^s:  First  origin,  6}"  to  the 
left  and  3f"  above  the  center  of  the  sheet;  second  origin,  IJ"  to 
the  right  and  6"  above  the  center ;  third  origin,  Sj"  to  the  right  and 
5 J"  above  the  center;  fourth  origin,  GV  to  the  left  and  4J"  below 
the  center ;  fifth  origin,  7}"  to  the  right  of  the  center  of  the  sheet 
on  the  horizontal  center  line. 

First  Origin:    Sphere  and  Cylinder. 

Draw  a  sphere  pierced  by  a  right  circular  cylinder.  The  center 
of  the  sphere  is  at  (2",  2",  2"),  its  diameter  sV'.  The  axis  of  the 
cylinder  is  P  (2",14",i"),  P'  (2",  1^,31").  Its  diameter  is  H". 
Praw  the  sphere  and  cylinder  in  fi,  V  and  S,  and  determine  the 
line  of  intersection  by  passing  planes  parallel  to  V  at  distances  of 
i",  r,  n"  and  If". 

Second  Origin:     Forked  End  of  Connecting-Rod. 

The  forked  end  of  a  connecting  rod  has  the  shape  of  a  surface  of 
revolution,  faced  off  at  the  sides  to  a  width  of  1^'',  as  shown  in  Fig. 
136.  The  centers  a,  I,  and  c  are  points  (2'',  1",  0"),  (2",  0",  f"), 
and  (2",  0",  1").  The  arc  which  has  cZ  as  a  center  is  tangent  at  its 
ends  to  the  adjacent  arc  and  to  the  side  of  the  1"  cylinder. 

Determine  the  continuation  of  the  line  of  intersection  of  the 
plane  and  surface  at  w,  by  passing  planes  parallel  to  fi  at  distances 
from  H  of  2i",  2f ",  ^",  2f "  and  2%".    Draw  no  side  view. 


Set  of  Descriptive  Drawings 


155 


Third  Origin:    Stub  End  of  Connecting-Rod. 

The  stub  end  of  a  connecting-rod  is  a  surface  of  revolution  faced 
off  at  the  sides  to  a  width  of  IJ",  and  pierced  by  bore-holes  parallel 
to  its  axis  as  shown  in  Fig.  137.     Centers  are  at  a  (IJ",  1",  0"), 

h  {r,  r,  0"),  c  (r,  r,  o"),  d  (sr,  o^  if),  and  e  (r,  0",  ir). 

Determine  the  continuation  of  the  line  of  intersection  at  lu  by 
passing  planes  parallel  to  H   at  distances  from  ff  of  lyV"^  ^¥'> 


Fig.  136. 


Fig.  137. 


1^",  and  If".  Draw  also  the  side  view  and  determine  the  ap- 
pearance of  the  edge  marked  u,  where  the  large  part  of  the  bore-hole 
intersects  the  surface  of  revolution,  by  means  of  the  same  system  of 
planes. 

Fourth  Origin:    Right  Circular  Cylinder  and  Cone. 

A  right  circular  cone  is  pierced  by  a  right  circular  cylinder,  the 
axes  intersecting  at  right  angles,  as  in  Fig.  62,  Art.  72.    The  axis 


156  Engineering  Descriptive  Geometry 

of  the  cone  is  P  (2f',  2^",  \"),  P'  (24",  2^",  2J").  The  base,  in  a 
plane  parallel  to  iHI,  is  a  circle  of  3}"  diameter.  The  axis  of  the 
cylinder  is  Q  {l\  2^",  If"),  Q'  (4",  2^",  If"),  and  its  diameter  is 

Draw  three  views  of  the  figures,  determining  the  line  of  inter- 
section by  planes  parallel  to  H.  It  is  best  not  to  pass  these  planes  at 
equal  intervals,  but  through  points  at  equal  angles  on  the  base  of 
the  cylinder.  Divide  the  base  of  the  cylinder  in  S  ii^to  arcs  of  30°, 
and  in  numbering  the  points  let  that  corresponding  to  F,  in  Fig.  62, 
be  numbered  0  and  let  H  be  numbered  6.  Insert  intermediate 
points  from  1  to  5  on  both  sides,  so  that  the  horizontal  planes  used 
for  the  determination  of  the  curve  of  intersection  are  seven  in 
number,  the  lowest  passing  through  the  point  0,  the  second  through 
the  two  points  1,  the  third  through  the  two  points  2,  etc.  Determine 
the  curve  of  intersection  by  these  planes. 

Draw  the  development  of  the  surface  of  the  cylinder,  cutting  the 
surface  on  the  element  00'  (or  FF'  in  Fig.  62).  Place  this  line  of 
the  development  vertically  on  the  sheet,  the  point  0  being  1"  to  the 
left  and  7^"  below  the  center  of  the  sheet,  and  0'  being  1"  to  the 
left  and  3}"  below  the  center  of  the  sheet. 

Draw  the  development  of  the  surface  of  the  cone  Note  that  the 
radius  of  the  base,  the  altitude,  and  the  slant  height  are  in  the 
ratio  of  3:4:5.  To  get  equally  spaced  elements  on  the  surface  of 
the  cone,  divide  the  arc  corresponding  to  BC  in  H,  Fig.  62,  into 
five  equal  spaces.  Number  the  point  B  0  and  C  5,  and  the  inter- 
mediate points  in  series.  Since  the  cone  is  symmetrical  about  two 
axes  at  right  angles,  one  quadrant  may  represent  all  four  quadrants. 
Put  the  vertex  of  the  developed  surface  3"  to  the  left  of  the  center 
of  the  sheet  and  1"  below  it,  and  consider  it  cut  on  the  line  PO  or 
PB.*  Locate  the  point  0  8|"  to  the  left  of  the  center  of  the  sheet  and 
1"  below  it.  Divide  the  development  into  four  quadrants  and  then 
divide  each  quadrant  into  five  parts,  numbering  the  21  points 
0,  1,  2,  3,  4,  5,  4,  3,  2,  1,  0,  1,  2,  3,  4,  5,  4,  3,  2,  1,  0. 

Fifth  Origin:    Cone  and  Double  Ogival  Point. 

In  this  figure  a  right  circular  cone  pierces  a  double  ogival  point. 
The  cone  has  a  vertical  axis,  PP',  the  vertex  P  being  at  (3",  IJ", 


Set  of  Descriptive  Drawings  157 

I"),  and  F,  the  center  of  the  base,  at  (3",  IJ'',  3^").    The  base  is  a 
circle  of  2^"  diameter  lying  in  a  horizontal  plane. 

The  ogival  point  has  an  axis  of  revolution,  Q  (|",  1^",  2"), 
Q'  (5f^  li",  2"),  5J"  long.  The  generating  line  is  an  arc  of  4" 
radius  of  which  QQ'  is  the  chord,  and  in  its  initial  position  the  arc 
has  its  center  at  (3",  IJ",  5'.'02).  Draw  three  views  of  the  cone 
piercing  the  double  ogival  surface,  and  determine  the  line  of  inter- 
section by  means  of  three  auxiliary  cutting  spheres,  centered  at  p, 
the  intersection  of  PP'  and  QQ'.  Use  diameters  of  2J",  2^",  and 
2iV"-    This  curve  appears  on  the  U.  S.  Navy  standard  3"  valve. 

General  Directions  for  Completing  the  Sheet. 

In  inking  the  sheet  show  one  cutting  plane  or  sphere  for  the 
determination  of  each  line  of  intersection,  and  show  clearly  how 
one  point  is  determined  in  each  view  of  each  figure.  Shade  the 
figures,  except  the  developments. 

In  the  legend  space  record  the  following  legend : 

SHEET    III.  (Block  letters  15/32"  high.) 

descriptive  geometry.  (All  caps  3/16"  high.)  ^ 

INTERSECTIONS  OF  SURFACES  OF  ^    '' 

REVOLUTION.  (All  caps  9/32"  high.) 

Name    (signature).  ClaSS.  (Caps  1/8"  high,  lower  case  1/12"  high.) 

Date.  (Caps  1/8"  high,  lower  case  1/12"  high.) 


SHEET  IV:     CONES,  ANCHOR  RING  AND  HEnCOIDAL 
SURFACES. 

Lay  out  center  lines,  cutting,  border,  working  lines,  and  legend 
space  as  before. 

From  the  center  of  the  sheet  plot  origins  as  follows:  First 
origin,  3^'  to  the  left  of  the  center  and  Syl"  above  the  center; 
second  origin,  5 J"  to  the  right  of  the  center  and  3"  above  the  center ; 
third  origin,  lOJ"  to  the  right  of  the  center  and  4 J"  above  the  center ; 
fifth  origin,  3"  to  the  right  of  the  center  and  6"  below  the  center. 


158  Engineering  Descriptive  Geometry 

First  Origin:    Intersecting  Inclined  Cones. 

Draw  two  intersecting  inclined  cones.  The  first  cone  has  its 
vertex  at  P  (1",  If,  i").  and  the  center  of  its  base  at  P'  {2",\\ 
11",  4|").  The  base  is  a  circle  of  3f"  diameter,  lying  in  a  plane 
parallel  to  fi-  The  second  cone  has  its  vertex  at  Q  (5",  1^",  2''46), 
and  the  center  of  its  base  at  Q'  (J",  IJ",  3J").  The  base  is  a  circle 
of  3"  diameter  lying  in  a  plane  parallel  to  g.  Draw  plan,  front 
elevation,  side  elevation,  and  an  auxiliary  projection  on  a  plane  JJ^ 
perpendicular  to  the  line  PQ,  the  trace  of  U  on  V^  passing  through 
the  point  M  (TJ",  0",  0").  Determine  the  line  of  intersection  of  the 
cones  by  auxiliary  cutting  planes  containing  the  line  PQ,  and  treat 
the  problem  on  the  supposition  that  the  cone  PP'  pierces  the  cone 

Second  Origin:    Helicoidal  Surface  for  Screw  Propeller. 

A  right  vertical  cylinder,  IJ"  in  diameter,  has  for  its  axis 
P  (2^',  n",  \"),  P'  (2i",  21",  3f ').  Projecting  from  the  cylinder 
is  a  line  A  (3i",  ^'\  -i"),  B  (^",  2i",  ^").  This  line,  moving 
uniformly  along  the  cylinder,  and  about  it  clockwise,  describes  one 
complete  turn  of  a  helicoidal  surface  of  3"  pitch.  Draw  plan  and 
front  elevation  of  the  figure.  This  helicoid  is  intersected  by  an 
elliptical  cylinder  of  which  the  generating  line  is  perpendicular  to 
H  and  the  directrix  is  an  ellipse  lying  in  H,  having  its  major  axis 
C  (2",  2V',  0"),  D  {Y,  24",  0"),  and  minor  axis  E  {1\",  3",  0"), 
F  {1\",  2",  0").  Find  the  intersection  of  the  two  surfaces.  Ink  in 
full  lines  only  the  circular  cylinder  and  the  intersection.  This 
portion  of  a  helicoidal  surface  is  similar  to  that  which  is  used  for 
the  acting  surface  of  the  ordinary  marine  screw  propeller,  of  3  or 
4  blades. 

Third  Origin:    Worm  Thread  Surface. 

A  worm  shaft  is  a  right  cylinder,  IJ"  in  diameter,  its  axis  being 
■P  (If",  IJ",  i"),  P'  (IJ",  If",  8f").  A  triple  right-hand  worm 
thread,  of  the  same  profile  as  in  Fig.  70,  projects  from  the  cylinder 
along  the  middle  6"  of,  its  length.    The  pitch  of  the  thread  is  4J", 


Set  of  Descriptive  Drawings  159 

so  that  each  thread  has  more  than  a  complete  turn.  The  outside 
diameter  of  the  worm  is  3".  Make  a  complete  drawing  of  the  plan 
and  front  elevation,  as  in  Fig.  70,  letting  the  worm  thread  begin  at 
any  point  on  the  circumference. 

Fourth  Origin:    Anchor  Ring  and  Planes. 

An  anchor  ring,  R,  is  formed  by  revolving  a  circle  of  1  J"  diameter, 
lying  in  a  plane  parallel  to  V  and  with  its  center  at  A  (1^",  2f", 
J"),  about  an  axis  perpendicular  to  W  and  piercing  Cil  at  the  point 
B  (^f":>  ^f'?  0")-  I^raw  plan,  front  elevation,  right  side  elevation 
(to  the  right  of  IHI),  and  left  side  elevation  (to  the  left  of  H)  on  a 
plane  S',  4J"  from  §.  A  plane  P,  parallel  to  S  at  f "  from  S,  cuts 
the  ring.  Draw  the  trace  of  P  on  f\,  and^the  intersection  of  P  and 
the  ring  on  S-  A  second  plane  P',  parallel  to  S  at  IJ"  distance, 
cuts  the  ring.  Draw  the  trace  P'H  and  the  intersection  P'R  on  S- 
A  third  plane  Q  is  parallel  to  V  at  If"  distance  from  V-  Draw  the 
trace  QTI  and  the  intersection  QR  on  V-  A  fourth  plane,  Q',  is 
parallel  to  V  at  2"  distance.  Draw  the  trace  Q'H  and  the  inter- 
section Q'R  on  V-  An  inclined  plane  T  is  perpendicular  to  S  and 
S',  its  trace  on  §'  passing  through  the  point  C  (4f'',  2f",  J"),  and 
inclining  down  to  the  right  at  such  an  angle  as  to  be  tangent  to  the 
projection  on  §'  of  the  generating  circle  when  its  center  is  at 
D  (2f",  li'\  i").  Draw  the  trace  of  T  on  §',  and  the  intersection 
TR  on  W.  Find  the  true  shape  of  TR  by  means  of  an  auxiliary 
plane  of  projection  \]  perpendicular  to  S',  cutting  §'  in  a  trace 
parallel  to  T8'  through  the  point  on  S'  whose  coordinates  are 
E  (4r,  0",  IJ")- 

•     General  Directions  for  Completing  the  Sheet. 

Ink  the  sheet  uniform  with  the  preceding  sheets,  and  in  the 
legend  space  record  the  following  legend : 

SHEET    ly.  (Block  letters  15/32"  high.) 

descriptive   geometry.  (All  caps  3/16"  high.) 

CONES,  ANCHOR  RING  AND  HELICOIDS.  (All  caps  9/32"  high.) 

Name  (signature).  Class.  (Caps  1/8"  high,  lower  case  1/12"  high.) 

Date.  (Caps  1/8"  high,  lower  case  1/12"  high.) 


Short-title  Catalogue 

OF    THE 

PUBLICATIONS 

OF 

JOHN  WILEY  &  SONS 

New  York 
London:  CHAPMAN    &  HALL,  Limited 


ARRANGED   UNDER  SUBJECTS 


Descriptive  circulars  sent  on  application.     Books  marked  with  an  asterisk  (*)  are 
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1 


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2 


4 

00 

1 

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3 

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2  50 

1 

50 

3  00 

1 

50 

1 

50 

4 

00 

2 

00 

2 

00 

5 

00 

5 

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4 

00 

5 

00 

7 

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2 

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4 

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5 

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1 

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1 

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1 

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1 

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1 

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1 

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3 

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3 

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6 

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6 

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3 

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5 

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5 

50 

1 

50 

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3 


3 

00 

3 

00 

3 

00 

1 

00 

1 

00 

2 

50 

3 

00 

4 

00 

3 

00 

1 

50 

1 

50 

2 

50 

3 

50 

3 

00 

4 

00 

3 

00 

1 

00 

2 

50 

3 

00 

5 

00 

2 

00 

$3  oa 

3  00 

1  50 

2  50 

1  25 

3  00 

1  00 

1  50 

3  00 

3  00 

2  00 

3  00 

1  25 

2  00 

4  00 

3  00 

4  00 

1  25 

2  00 

1  50 

2  00 

5  00 

),3  00 

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9 


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MacCord's  Elements  of  Descriptive  Geometry Svo,  3  00 

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Robinson's  Principles  of  Mechanism Svo,  3  00 

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10 


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Schapper's  Laboratory  Guide  for  Students  in  Physical  Chemistry 12mo,  1  00 

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11 


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Elements  of  Geometry Svo,  1  75 

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No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
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Variable Svo.  2  00 

12 


*  Waterbury's  Vest  Pocket  Hand-book  of  Mathematics  for  Engineers. 

2JX6I  inches,  mor.  $1  00 

*  Enlarged  Edition,  Including  Tables mor.     1  60 

Weld's  Determinants 8vo,     1  00 

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MATERIALS  OP   ENGINEERING,  STEAM-ENGINES   AND   BOILERS. 

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*  "                   "                 "       Abridged  Ed Svo,  150 

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Carpenter's  Experimental  Engineering Svo,  6  00 

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Compton's  First  Lessons  in  Metal  Working 12mo,  1  50 

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Coolidge's  Manual  of  Drawing Svo,  paper,  1  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
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Cromwell's  Treatise  on  Belts  and  Pulleys 12mo,  1  50 

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Flather's  Dynamometers  and  the  Measurement  of  Power 12mo,  3  00 

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Hering's  Ready  Reference  Tables  (Conversion  Factors) 16mo,  mor.  2  50 

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Hutton's  Gas  Engine Svo,  6  00 

Jamison's  Advanced  Mechanical  Drawing Svo,  2  00 

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Jones's  Gas  Engine Svo,  4  00 

Machine  Design: 

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Mechanical  Drawing 4to,  4  00 

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Reid's  Course  in  Mechanical  Drawing Svo,  2  00 

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13 


51 

50 

3 

00 

3 

00 

3 

00 

3 

00 

3 

00 

3 

50 

1 

00 

3 

00 

1 

50 

1 

25 

7 

50 

1 

00 

1 

50 

5 

00 

5 

00 

2 

50 

Richards's  Compressed  Air 12mo, 

Robinson's  Principles  of  Mechanism 8vo, 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo, 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo, 

Smith's  (O.)  Press-working  of  Metals 8vo, 

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2|X5|  inches,  mor. 

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Weisbach's    Kinematics   and    the    Power   of   Transmission.     (Herrmann — 

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Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

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14 


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Entropy  Table Svo,     1  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines.  .  .  .  Svo,     5  00 

Valve-gears  for  Steam-engines Svo,     2  50 

Peabody  and  Miller's  Steam-boilers Svo,     4  00 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) 12mo,     1  25 

Reagan's  Locomotives:  Simple,  Compound,  and  Electric.     New  Edition. 

Large  12mo,     3  50 

Sinclair's  Locomotive  Engine  Running  and  Management 12mo, 

Smart's  Handbook  of  Engineering  Laboratory  Practice 12mo, 

Snow's  Steam-boiler  Practice Svo, 

Spangler's  Notes  on  Thermodynamics 12mo, 

Valve-gears Svo, 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering Svo, 

Thomas's  Steam-turbines Svo, 

Thurston's  Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indi- 
cator and  the  Prony  Brake Svo, 

Handy  Tables Svo, 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation  Svo, 

Manual  of  the  Steam-engine 2  vols.,  Svo, 

Part  I.      History,  Structure,  and  Theory Svo, 

Part  II.      Design,  Construction,  and  Operation Svo, 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-water.     (Patterson.) 
.  Svo, 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) Svo, 

Whitham's  Steam-engine  Design Svo, 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .Svo, 


MECHANICS    PURE   AND    APPLIED. 

Church's  Mechanics  of  Engineering Svo,  6  00 

Notes  and  Examples  in  Mechanics Svo,  2  00 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools  .12mo,  1  50 
Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.     I.      Kinematics Svo  3  50 

Vol.  II.     Statics Svo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  00 

*  Greene's  Structural  Mechanics Svo,  2  50 

Hartmann's  Elementary  Mechanics  for  Engineering  Students.      (In  Press.) 
James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Large  12mo,  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics 12mo,  3  00 

Lanza's  Applied  Mechanics Svo,  7  50 

*  Martin's  Text  Book  on  Mechanics,  Vol.  I,  Statics 12mo,  1  25 

*  Vol.  II,  Kinematics  and  Kinetics.  12mo,  1  50 

Maurer's  Technical  Mechanics Svo,  4  00 

*  Merriman's  Elements  of  Mechanics 12mo,  1  00 

Mechanics  of  Materials Svo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics Svo,  4  00 

15 


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Robinson's  Principles  of  Mechanism 8vo,  $3  00' 

Sanborn's  Mechanics  Problems Large  12mo,      1   5©' 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,     3  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,     3  00 

Principles  of  Elementary  Mechanics 12mo,     1   25 


MEDICAL. 

*  Abderhalden's  Physiological   Chemistry   in   Thirty   Lectures.      (Hall   and 

Defren.) Svo, 

von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) 12mo, 

Bolduan's  Immune  Sera 12mo, 

Bordet's  Studies  in  Immunity.      (Gay.) Svo, 

Chapin's  The  Sources  and  Modes  of  Infection.      (In  Press.) 
Davenport's  Statistical  Methods  with  Special  Reference  to  Biological  Varia- 
tions  16mo,  mor. 

Ehrlich's  Collected  Studies  on  Immunity.      (Bolduan.) Svo, 

*  Fischer's  Physiology  of  Alimentation Large  12mo, 

de  Fursac's  Manual  of  Psychiatry.      (Rosanoff  and  Collins.)..  .  .Large  12mo, 

Hammarsten's  Text-book  on  Physiological  Chemistry.      (Mandel.) Svo, 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .Svo, 

Lassar-Cohn's  Practical  Urinary  Analysis.      (Lorenz.) 12mo, 

Handel's  Hand-book  for  the  Bio-Chemical  Laboratory 12mo. 

*  Nelson's  Analysis  of  Drugs  and  Medicines 12mo. 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.      (Fischer.)  ..12mo, 

*  Pozzi-Escot's  Toxins  and  Venoms  and  their  Antibodies.      (Cohn.).  .  12nio, 

Rostoski's  Serum  Diagnosis.      (Bolduan.) 12nio, 

Ruddiman's  Incompatibilities  in  Prescriptions Svo, 

Whys  in  Pharmacy I2mo, 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Omdorflf.)  ..  ..Svo, 

*  Satterlee's  Outlines  of  Human  Embryology 12mo, 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students Svo, 

*  Whipple's  Tyhpoid  Fever Large  12mo, 

*  Woodhull's  Military  Hygiene  for  Officers  of  the  Line Large  12mo, 

*  Personal  Hygiene 12mo, 

Worcester  and  Atkinson's  Small  Hospitals  Establishment  and  Maintenance, 
and  Suggestions  for  Hospital  Architecture,  *vith  Plans  for  a  Small 
Hospital 12mo,     1  25 

METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis Svo,  4  00' 

Bolland's  Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  used 

in  the  Practice  of  Moulding 12mo,  3  OO' 

Iron  Founder 12mo,  2  50 

Supplement 12mo,  2  50 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  3  00> 

*  Iles's  Lead-smelting 12mo,  2  50 

Johnson's    Rapid    Methods   for    the  Chemical   Analysis   of   Special    Steels, 

Steel-making  Alloys  and  Graphite Large  12mo,  3  00* 

Keep's  Cast  Iron Svo,  2  50- 

Le  Chatelier's  High-temperature  Measurements.     (Boudouard — Burgess.) 

12mo,  3  00 

Metcalf's  Steel.     A  Manual  for  Steel-users 12mo.  2  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.).  .  12mo,  2  50- 

*  Ruer's  Elements  of  Metallography.      (Mathewson.) Svo,  3  00- 

Smith's  Materials  of  Machines 12mo,  1  00' 

Tate  and  Stone's  Foundry  Practice 12mo,  2  00' 

Thurston's  Materials  of  Engineering.     In  Three  Parts Svo,  S  00 

Part  I.      Non-metallic  Materials  of  Engineering,  see  Civil  Engineering, 

page  9. 

Part  II.    Iron  and  Steel Svo,     3  50 

Part  III.  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,     2  50> 

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5  00 

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Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  $3  00' 

West's  American  Foundry  Practice 12mo,     2  5Q 

Moulders'  Text  Book 12mo.     2  50 


MINERALOGY. 

Baskerville's  Chemical  Elements.      (In  Preparation.) 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo, 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo, 

Butler's  Pocket  Hand-book  of  Minerals 16mo,  mor. 

Chester's  Catalogue  of  Minerals Svo,  paper, 

Cloth, 

P  Crane's  Gold  and  Silver .  Svo, 

Dana's  First  Appendix  to  Dana's  New  "System  of  Mineralogy".  .Large  Svo, 
Dana's  Second  Appendix  to  Dana's  New  "  System  of  Mineralogy." 

Large  Svo, 

Manual  of  Mineralogy  and  Petrography 12mo, 

Minerals  and  How  to  Study  Them 12mo, 

System  of  Mineralogy Large  Svo,  half  leather, 

Text-book  of  Mineralogy Svo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Eakle's  Mineral  Tables Svo, 

Eckel's  Stone  and  Clay  Products  Used  in  Engineering.      (In  Preparation.) 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

Groth's  The  Optical  Properties  of  Crystals.     (Jackson.)      (In  Press.) 
Groth's  Introduction  to  Chemical  Crystallography  (Marshall) 12mo, 

*  Hayes's  Handbook  for  Field  Geologists 16mo,  mor. 

Iddings's  Igneous  Rocks Svo, 

Rock  Minerals Svo, 

Johannsen's  Determination  of  Rock-forming  Minerals  in  Thin  Sections.  Svo, 

With  Thumb  Index     5  00 

*  Martin's  Laboratory     Guide    to    Qualitative    Analysis    with    the    Blow- 

pipe  12mo,         60 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses Svo, 

Stones  for  Building  and  Decoration Svo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

Svo,  paper. 

Tables  of  Minerals,    Including  the  Use  of  Minerals  and  Statistics  of 

Domestic  Production Svo, 

*  Pirsson's  Rocks  and  Rock  Minerals 12mo, 

*  Richards's  Synopsis  of  Mineral  Characters 12mo,  mor. 

*  Ries's  Clays:  Their  Occurrence.  Properties  and  Uses Svo, 

*  Ries  and  Leighton's  History  of  the  Clay-working  Industry  of  the  United 

States Svo. 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks Svo, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks Svo, 


MINING. 

*  Beard's  Mine  Gases  and  Explosions Large  12mo, 

*  Crane's  Gold  and  Silver Svo, 

*  Index  of  Mining  Engineering  Literature Svo. 

*  Svo,  mor. 

Ore  Mining  Methods.      (In  Press.) 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Eissler's  Modem  High  Explosives Svo, 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

Ihlseng's  Manual  of  Mining Svo, 

*  Iles's  Lead  Smelting 12mo, 

Peele's  Compressed  Air  Plant  for  Mines Svo, 

Riemer's  Shaft  Sinking  Under  Difficttlt  Conditions.     (Coming  and  Peele.)Svo, 

*  Weaver's  Military  Explosive*. Svo, 

Wilson's  Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 12mo, 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12mo, 

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1  25. 

SANITARY    SCIENCE. 

Association  of  State  and  National  Food  and  Dairy  Departments,  Hartford 

Meeting,  1906 8vo,  $3  00 

Jamestown  Meeting,  1907 8vo,  3  00 

*  Bashore's  Outlines  of  Practical  Sanitation 12mo,  1  25 

Sanitation  of  a  Country  House 12mo,  1  00 

Sanitation  of  Recreation  Camps  and  Parks 12mo,  1  00 

Chapin's  The  Sources  and  Modes  of  Infection.      (In  Press.) 

Folwell's  Sewerage.      (Designing,  Construction,  and  Maintenance.) 8vo,  3  00 

Water-supply  Engineering.  . 8vo,  4  00 

Fowler's  Sewage  Works  Analyses. 12mo,  2  00 

Fuertes's  Water-filtration  Works 12mo,  2  50 

Water  and  Public  Health 12mo,  1  50 

Gerhard's  Guide  to  Sanitary  Inspections 12mo,  1  50 

*  Modern  Baths  and  Bath  Houses 8vo,  3  00 

Sanitation  of  Public  Buildings 12mo,  1  50 

*  The  Water  Supply,  Sewerage,  and  Plumbing  of  Modern  City  Buildings. 

8vo,  4  00 

Hazen's  Clean  Water  and  How  to  Get  It Large  12mo,  1  50 

Filtration  of  Public  Water-supplies 8vo,  3  00 

Kinnicut,  Winslow  and  Pratt's  Purification  of  Sewage.      (In  Preparation.) 
Leach's  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Examination  of  Water.      (Chemical  and  Bacteriological) 12mo,  1  25 

Water-supply.      (Considered  principally  from  a  Sanitary  Standpoint). 

8vo,  4  00 

*  Merriman's  Elements  of  Sanitary  Engineering 8vo,  2  00 

Ogden's  Sewer  Construction 8vo,  3  00 

Sewer  Design 12mo,  2  00 

Parsons's  Disposal  of  Municipal  Refuse 8vo,  2  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis 12mo,  1  50 

*  Price's  Handbook  on  Sanitation 12mo,  1  50 

Richards's  Cost  of  Cleanness 12mo,  1  00 

Cost  of  Food.      A  Study  in  Dietaries 12mo,  1  00 

Cost  of  Living  as  Modified  by  Sanitary  Science 12mo,  1  00 

Cost  of  Shelter 12mo,  1  00 

*  Richards  and  Williams's  Dietary  Computer 8vo,  1  50 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  00 

*  Richey's     Plumbers',      Steam-fitters',    and     Tinners'     Edition     (Building 

Mechanics'  Ready  Reference  Series) 16mo,  mor.  1  50 

Rideal's  Disinfection  and  the  Preservation  of  Food 8vo,  4  00 

Sewage  and  Bacterial  Purification  of  Sewage 8vo,  4  00 

Soper's  Air  and  Ventilation  of  Subways 12mo,  2  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Venable's  Garbage  Crematories  in  America 8vo,  2  00 

Method  and  Devices  for  Bacterial  Treatment  of  Sewage 8vo,  3  00 

Ward  and  Whipple's  Freshwater  Biology.      (In  Press.) 

Whipple's  Microscopy  of  Drinking-water Svo,  3  50 

*  Typhoid  Fever Large  12mo,  3  00 

Value  of  Pure  Water Large  12mo,  1  00 

Winslow's  Systematic  Relationship  of  the  Coccaceae Large  12mo,  2  50 


MISCELLANEOUS. 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo  1   50 

Ferrel's  Pooular  Treatise  on  the  Winds , Svo,  4  00 

Fitzgerald's  Boston  Machinist 18mo,  1   00 

Gannett's  Statistical  Abstract  of  the  World 24mo.  75 

Haines's  American  Railway  Management 12mo,  2  50 

Hanausek's  The  Microscopy  of  Technical  Products.      (Winton) 8vo,  5  00 

18 


Jacobs's  Betterment    Briefs.     A    Collection    of    Published    Papers    on    Or- 
ganized Industrial  Efficiency 8vo,  $3  50 

Metcalfe's  Cost  of  Manufactures,  and  the  Administration  of  Workshops.. 8vo,  5  00 

Putnam's  Nautical  Charts 8vo,  2  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute  1824-1894. 

Large  12mo,  3  00 

Rotherham's  Emphasised  New  Testament Large  8vo,  2  00 

Rust's  Ex-Meridian  Altitude,  Azimuth  and  Star-finding  Tables 8vo,  5  00 

Standage's  Decoration  of  Wood,  Glass,  Metal,  etc 12mo,  2  00 

Thome's  Structural  and  Physiological  Botany.     (Bennett) 16mo,  -2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider) 8vo,  ?  00 

Winslow's  Elements  of  Applied  Microscopy 12mo,  1  50 


HEBREW   AND    CHALDEE    TEXT-BOOOKS. 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to.  half  mor,     5  00 

Green's  Elementary  Hebrew  Grammar 12mo,     1  25 


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UNIVERSITY  OF  CALIFORNIA  UBRARY 


